2014-08-10

Stability of complete minimal Lagrangian submanifold and L^2 harmonic 1-formsWe show that a non-compact complete stable minimal Lagrangian submanifold L in a Kahler manifold with positive Ricci curvature has no non-trivial L2 harmonic 1-forms. This is an extension of Y.G. Oh’s result on the compact submanifold case. We report several important related results in this field. This is a joint work with my PhD student S. Ueki.

Generalizations of the Catenoid and the Helicoid

We will begin by discussing quasi-umbilicty and conformal flatness; in the study of conformally flat, minimal submanifolds of Euclidean space we look at this as a generalization of a surface of revolution. This leads to the generalized catenoid; in the hypersurface case this has been known since 1975 but we will also discuss a more recent result in higher codimension.
In 1999 Castro and Urbano introduced a Lagrangian catenoid; we will review their result and study conformally flat, minimal, Lagrangian submanifolds in Cn.
Turning to helicoids we will first mention a result of J. R. Vanstone and the author on complete, minimal hypersurfaces of Euclidean space admitting a codimension 1 foliation by Euclidean spaces leading to a generalized helicoid, but which is not locally isometric to the generalized catenoid. Then we discuss Lagrangian subman- ifolds in Cn foliated by (n − 1)-planes and especially in C2 leading to a Lagrangian helicoid in C2. We also show that there exists a 1-parameter family of ruled La- grangian surfaces connecting this Lagrangian helicoid to a Lagrangian catenoid.

Canonical forms under certain actions on the classical simple Lie groups

A maximal torus of a compact connected Lie group can be seen as a canonical form of adjoint action since any two maximal tori can be transformed each other by an inner automorphism. A. Kollross defined a sigma-action on a compact Lie group which is a generalization of the adjoint action. Since a sigma-action is hyperpolar, it has a canonical form called a section. In this talk we study the structure of the orbit space of a sigma-action and properties of each orbit, such as minimal, austere and totally geodesic, using symmetric triads introduced by the author, when sigma is an involution of outer type on the compact simple Lie groups of classical type. As an application, we investigate the fixed point set of a holomorphic isometry of an irreducible Hermitian symmetric space of compact type which does not belong to the identity component of the group of holomorphic isometries.

Totally geodesic surfaces of Riemannian symmetric spaces

I will talk about the classification of non-flat totally geodesic surfaces in irreducible Riemannian symmetric spaces G/K where G is one of the compact classical Lie group.
Let G be a compact simple Lie group and θ be an involutive automorphism ofG. WedenotebygtheLiealgebraofGandputk={X∈g:θ(X)=X}, m = {X ∈ g : θ(X) = −X}. The subspace m is identified with the tangent space of G/K at the origin o = eK. A subspace s of m with the property
[s, [s, s]] ⊂ s
is called a Lie triple system. There exists a one-to-one correspondence between the set of totally geodesic submanifolds of M and the set of Lie triple systems.
Let G be one of SO(n), SU(n) or Sp(n) and S be a non-flat totally geodesic surface of G/K. The universal covering group U of the isometry group of S is iso- morphictoSU(2). LetV beCn ifG=SU(n)orSO(n)orC2n ifG=Sp(n)and consider the action of U on V through the homomorphism U → G. Our classifica- tion depends on the study of the relation between the weight space decomposition of V under action of U and the decomposistion of V by θ.

Orbifold holomorphic discs and crepant resolutions

This is a note of a lecture at the conference, “Real and Complex submanifolds”. We survey the definition and properties of orbifold holomorphic discs and an ap- plication to crepant resolution conjecture.

Floer homology for the Gelfand-Cetlin system

In this talk we discuss Lagrangian intersection Floer theory for the Gelfand-Cetlin system, which is a completely integrable system on a flag manifold of type A in- troduced by Guillemin and Sternberg [2]. Lagrangian intersection Floer theory for torus orbits in a toric manifold have developed by Fukaya, Oh, Ohta and Ono [1]. Using a toric degeneration of a flag manifold, we compute the potential function in the sense of Fukaya-Oh-Ohta-Ono, and show that it coincides with the super- potential of the Landau-Ginzburg mirror of the flag manifold ([3]). We also study Floer homology for non-torus Lagrangian fibers, which appear on the boundary of the momentum polytope of the Gelfand-Cetlin system.
References
[1] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Lagrangian Floer theory and mirror symmetry on compact toric manifolds, arXiv:1009.1648.
[2] V. Guillemin and S. Sternberg, The Gelfand-Cetlin system and quantization of the complex flag manifolds, J. Funct. Annal. 52 (1983), 106-128.
[3] T. Nishinou, Y. Nohara, and K. Ueda, Toric degenerations of Gelfand-Cetlin systems and potential functions, Adv. Math. 224 (2010), 648-706.

The fixed point set of a holomorphic isometry and the intersection of two real forms in the complex Grassmann manifold

In the previous papers [4], [5] and [6] the second and the third authors proved that the intersection of two real forms in a Hermitian symmetric space of compact type is an antipodal set by making use of Chen-Nagano theory. In this talk, first, we investigate the fixed point set of a holomorphic isometry of the complex Grassmann manifold which belongs to the identity component of the group of holomorphic isometries. Second, we investigate the intersection of two real forms which are congruent to the real Grassmann manifold in the complex Grassmann manifold, which is treated in [3]. Lastly, we refer to the relation between the intersection of such real forms and the fixed point set of a holomorphic isometry. We showed these results in more general situation, that is, we investigated the relation between the fixed point set of a holomorphic isometry in a Hermitian symmetric space of compact type and the intersecion of two real forms in a Hermitian symmetric space of compact type in [2] where we made use of symmetric triads introduced by the first author in [1], which gives an alternative proof of the fact that the intersection of two real forms is an antipodal set.
References
[1] O. Ikawa, The geometry of symmetric triad and orbit spaces of Hermann actions, J. Math. Soc. Japan, 63 (2011) 79–136.
[2] O. Ikawa, M. S. Tanaka and H. Tasaki, The fixed point set of a holomorphic isometry, the intersection of two real forms in a Hermitian symmetric space of compact type and symmetric triads, in preparation.
[3] H. Iriyeh, T. Sakai and H. Tasaki, On the structure of the intersection of real flag manifolds in a complex flag manifold, to appear in Advanced Studies in Pure Mathematics.
[4] M. S. Tanaka and H. Tasaki, The intersection of two real forms in Hermitian symmetric spaces of compact type, J. Math. Soc. Japan 64 (2012), 1297–1332.
[5] M. S. Tanaka and H. Tasaki, The intersection of two real forms in Hermitian symmetric spaces of compact type II, to appear in J. Math. Soc. Japan.
[6] M. S. Tanaka and H. Tasaki, Correction to: “The intersection of two real forms in Hermitian symmetric spaces of compact type”, preprint.

As SL2(C) topological invariant of knots

We show that the SL2(C) algebro-geometric invariant defined in [3] for knots is indeed an SL2(C) topological invariant. The main ingredient is our short geomet- ric proof of the coincidence of the algebro-geometric multiplicity and topological multiplicity of the intersection of curves on a smooth surface.
References
[1] S. Akbulut and J. McCarthy, Casson’s invariant for oriented homology 3-spheres. An expo- sition, Mathematical Notes, Vol. 36, Princeton University Press, Princeton, NJ., 1990.
[2] D. Cooper, M. Culler, H. Gillet, D.D. Long and P.B. Shalen, Plane curves associated to
character varieties of 3-manifolds. Invent. Math. Vol. 118, (1994), 47–74.
[3] W. Li and Q. Wang, An SL2(C) Algebro-Geometric Invariant of Knots. Internat. J. Math.
Vol. 22 (2011), no. 9, 1209–1230.
[4] J. P. Serre, Local Algebra, Springer Monographs in Mathematics, 2000.
[5] K. Walker, An extension of Casson’s invariant. Annals of Mathematics Studies, Vol. 126,
Princeton University Press, Princeton, NJ., 1992.

Some geometric aspects of the Hessian one equation

The Hessian one equation and its complex resolution provides an important tool in the study of improper affine spheres in R3 with some kind of singularities. The singular set can be characterized and, in most of the cases, it determines the surface. Here, we show how to obtain improper affine spheres with a prescribed singular set and construct some global examples with the desired singularities. We also classify improper affine spheres admitting a planar singular set.

Beyond generalized Sasakian-space-forms

In this talk we will review some recent advances on the theory of generalized Sasakian-space-forms, as well as some new directions in which this theory is being developed now.
References
[1] P. Alegre, D. E. Blair and A. Carriazo, Generalized Sasakian-space-forms, Israel J. Math. 141 (2004), 157-183.
[2] P. Alegre and A. Carriazo, Structures on generalized Sasakian-space-forms, Differential Geom. Appl. 26 (2008), 656-666.
[3] P. Alegre and A. Carriazo, Semi-Riemannian generalized Sasakian-space-forms, submitted.
[4] A. Carriazo, On generalized Sasakian-space-forms, Proceedings of the Ninth International
Workshop on Diff. Geom. 9 (2005), 31-39.
[5] A. Carriazo, L. M. Fern ́andez and Ana M. Fuentes, Generalized S-space-forms with two
structure vector fields, Advances in Geom. 10 (2010), 205-219.
[6] A. Carriazo and V. Mart ́ın-Molina, Almost cosymplectic and almost Kenmotsu (κ,μ,ν)-
spaces, Mediterr. J. Math. 10 (2013), 1551-1571.
[7] A. Carriazo, V. Mart ́ın-Molina and M. M. Tripathi, Generalized (κ, μ)-space forms, Mediterr.
J. Math. 10 (2013), 475-496.

Reeb flow invariant Ricci tensors

An almost contact three-manifold is a 3-dimensional differentiable manifold M whose structure group of the linear frame bundle is reducible to U (1) × {1}. Then we have a compatible Riemannian metric g on M. In this talk, we study almost contact metric three-manifolds whose Ricci operator S is invariant along the Reeb flow ξ. Mainly, we prove the following theorems.
Theorem 1.([1]) The Ricci operator S of a contact metric three-manifold M is invariant along the Reeb flow ξ, that is, M satisfies £ξS = 0 if and only if M is Sasakian or locally isometric to a Lie group SU(2), SL(2,R), E(2) (the group of rigid motions of Euclid 2-plane) with a left invariant contact Riemannian metric respectively.
Theorem 2.([2]) An almost cosymplectic three-manifold M satisfies £ξS = 0 if and only if M is cosymplectic or locally isometric to the group E(1,1) of rigid motions of Minkowski 2-space with a left invariant almost cosymplectic structure.
It should be remarkable that E(1,1) admits also a left invariant contact metric structure. But, it does not enjoy this symmetry property any more.
Theorem 3.([3]) An almost Kenmotsu three-manifold M satisfies £ξS = 0 if and only if M is locally isometric to either a hyperbolic space H3(−1) or a non- unimodular Lie group with a left invariant almost Kenmotsu structure.
We note that a non-unimodular Lie group appeared in the above Theorem 3 includes a product space H2(−4) × R.
References
[1] J. T. Cho, Contact 3-manifolds with the Reeb flow symmetry, Tohˆoku Math. J. (to appear). [2] J. T. Cho, Reeb flow symmetry on almost cosymplectic three-manifolds, preprint.
[3] J. T. Cho and M. Kimura, Reeb flow symmetry on almost contact three-manifolds, preprint.

A classification of Ricci solitons as (k,u)-contact metrics

In [2], Sharma studied Ricci soliton within the frame-work of contact geometry and proved that a complete K-contact gradient Ricci soliton is compact Einstein Sasakian. In [3], Sharma and Ghosh showed that a non-trivial 3-dimensional Sasakian Ricci soliton is homothetic to the standard Sasakian metric on nil3. Recently, Ghosh and Sharma [1] have shown in higher dimensions that a non-trivial Sasakian Ricci soliton is expanding and null η-Einstein (transverse Calabi-Yau). Such a characterization permits us to identify the Sasakian metric on the Heisenberg group H2n+1as an explicit example of non-trivial Ricci soliton of such type. In this paper we study (2n+1)-dimensional Ricci solitons (i) as non-Sasakian (k, μ)-contact metrics (a generalization of Sasakian metrics) and show that it is locally flat in dimension 3, and locally isometric to the trivial sphere bundle En+1 ×Sn(4) in higher dimensions; and (ii) as complete K-contact metric (another generalization of Sasakian metric) and assuming the generating vector field strict contact, show that it is compact Einstein Sasakian.
References
[1] Ghosh, A. and Sharma, R., Sasakian metric as a Ricci soliton and related results, J. Geom. Phys. 75 (2014), 1-6.
[2] Sharma, R., Certain results on K-contact and (k, μ)-contact manifolds, J. Geom. 89 (2008), 138-147.
[3] Sharma, R. and Ghosh, A., Sasakian 3-manifold as a Ricci soliton represents the Heisenberg group, Internat. J. Geom. Methods Mod. Phys. 8 (2011), 149-154.

Riemannian problems with a fundamental differential system

We introduce the reader to a fundamental exterior differential system of Riemannian geometry which arises naturally with every oriented Riemannian n+1-manifold
M. Such system is related to the well-known metric almost contact structure on the unit sphere tangent bundle SM, so we endeavor to include the theory in the field of contact systems. Our EDS is already known in dimensions 2 and 3, where it was used by Ph. Griffiths in applications to mechanical problems and Lagrangian systems. It is also known in any dimension but just for flat Euclidean space. Having found the Lagrangian forms αi ∈ Ωn, 0 ≤ i ≤ n, we are led to the associated functionals Fi(N) =

A new technique for the study of complete maximal hypersurfaces in certain open Generalized Robertson-Walker spacetimes

An (n + 1)-dimensional Generalized Robertson-Walker (GRW) spacetime [1] such that the universal Riemannian covering of the fiber is parabolic (thus so is the fiber) is said to be spatially parabolic. This class of spacetimes allows to model open relativistic universes which extend to the spatially closed GRW spacetimes from the viewpoint of the geometric-analysis of the fiber. On the contrary to spa- tially closed GRW spacetimes, these spacetimes could be compatible with certain cosmological principle [3]. We introduce here a new technique for the study of non- compact complete spacelike hypersurfaces in such spacetimes. Thus, a complete spacelike hypersurface in a spatially parabolic GRW spacetime inherits the parabol- icity, whenever some boundedness assumptions on the restriction of the warping function to the spacelike hypersurface and on the hyperbolic angle between the unit normal vector field and a certain timelike vector field are assumed. Conversely, the existence of a simply connected parabolic spacelike hypersurface, under the previ- ous assumptions, in a GRW spacetime also leads to its spatial parabolicity. Then, all the complete maximal hypersurfaces in a spatially parabolic GRW spacetime are determined in several cases, extending known uniqueness results [2]. Finally, all the entire solutions of the maximal hypersurface equation on a parabolic Riemannain manifold are found in several cases, solving new Calabi-Bernstein problems.
References
[1] L.J. Al ́ıas, A. Romero and M. Sa ́nchez, Uniqueness of complete spacelike hypersurfaces of constant mean curvature in Generalized Robertson-Walker spacetimes, Gen. Relat. Gravit., 27 (1995), 71–84.
[2] M. Caballero, A. Romero and R.M. Rubio, Constant mean curvature spacelike hypersurfaces in Lorentzian manifolds with a timelike gradient conformal vector field, Classical Quant. Grav., 28 (2011), 145009–145022.
[3] A. Romero R.M. Rubio and J.J. Salamanca, Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson-Walker spacetimes, Classical Quant. Grav., 30 (2013), 115007(1-13).

Information geometry of Barycenter map

Using barycenter of Busemann function we define a map, called barycenter map from a space P(∂X) of probability measures over an ideal boundary ∂X to an Hadamard manifold X, a non-compact, simply connected, complete Riemannian manifold of non-positive curvature. We show that the space P(∂X) carries a fibre space structure over X from a viewpoint of information geometry([3], [4]). Fol- lowing the idea of [1], [2] we present a theorem which states that under cetain hy- potheses of information geometry a homeomorphism Φ of ∂X induces, via the push- forward process for probability measures, an isometry of X whose ∂X-extension coincides with Φ.
References
[1] G. Besson, G. Courtois and S. Gallot, Entropies et Rigidit ́es des espaces localement sym ́etriques de courbure strictement n ́egative, Geom Func. Anal., 5. (1995), 731-799.
[2] E. Douady and C. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math., 157. (1986), 23-48.
[3] T. Friedrich, Die Fisher Information und symplektische Strukturen, Math. Nachr., 153. (1991), 273-296.
[4] M. Itoh and S. Satoh, The Fisher Information metric, Poisson kernels and harmonic maps, Diff. Geom. Appl., 29. (2011), S107-S115.

2014-08-11

Bonnesen-style symmetric mixed isoperimetric inequalityFor convex domains Ki (i = 0, 1) (compact convex sets with non-empty interiors) in the Euclidean plane R2, denote by Ai and Pi areas and circum-perimeters, respectively. The symmetric mixed isoperimetric deficit is
∆(K0, K1) := P02P12 − 16π2A0A1.
We introduce Bonnesen-style symmetric mixed inequalities, that is, inequalities of the form ∆(K0,K1) ≥ B(K0,K1), where B(K0,K1) is a non-negative invariant of geometric significance and vanishes if and only if both K0 and K1 are discs. We also derive some reverse Bonnesen-style symmetric mixed inequalities. Those inequalities are natural generalizations of known geometric inequalities, such as, the known classical isoperimetric inequality.
References
[1] Banchoff, T. F., Pohl, W. F., A generalization of the isoperimetric inequality, J. Diff. Geo. 6 (1971), 175-213.
[2] Blaschke, W., Vorlesungen u ̈ber Intergralgeometrie, 3rd ed. Deutsch. Verlag Wiss., Berlin, 1955.
[3] Bokowski, J., Heil, E., Integral representation of quermassintegrals and Bonnesen-style in- equalities, Arch. Math. 47 (1986), 79-89.
[4] Bonnesen, T., Les probl ́ems des isop ́erim ́etres et des is ́epiphanes, Paris, 1929.
[5] Bonnesen, T., Fenchel, W., Theorie der konvexen K ̈oeper, Berlin-Heidelberg-New York, 1934, 2nd Ed., 1974.
[6] Bottema, O., Eine obere Grenze fu ̈r das isoperimetrische Defizit ebener Kurven, Nederl. Akad. Wetensch. Proc. A66 (1933), 442-446.
[7] Burago, Yu. D., Zalgaller, V. A., Geometric Inequalities, Springer-Verlag Berlin Heidelberg, 1988.
[8] Croke, C., A sharp four-dimensional isoperimetric inequality, Comment. Math. Helv. 59 (1984), no. 2, 187-192.

Harmonic functions and parallel mean curvature surfaces

Minimal surfaces in a Euclidean three space are closely related to the complex func- tion theory. A constant mean curvature surface is constructed by a harmonic map- ping using the generalized Weierstrass representation formula. In this talk, I will show you a surface which is constructed by a harmonic function. It is immersed in a comlex two dimensional complex space form with parallel mean curvature vector. We prove that the Kaehler angle function of the surface is obtained by a functional transformation of a harmonic function. And, then, the 1st and 2nd fundamental forms of the surface are explicitly expressed by the Kaehler angle function. We also prove that, conversely, for a given harmonic function on a simply connected Riemann surface, there exists a parallel mean curvature surface in the complex projective plane (and also in the complex hyperbolic plane). As the byproduct, we show that any Riemann surface can be locally imbedded in the complex hyperbolic plane as a parallel mean curvature surface.

Real hypersurfaces in Kähler manifolds

We consider real hypersurfaces of compact K ̈ahler manifolds and show that real hypersurfaces of K ̈ahler manifolds induced by Morse functions have contact struc- tures. As examples we consider preimages of regular values of Morse functions on complex projective spaces, and real cosymplectic hypersurfaces of the products of K ̈ahler manifolds and torus.
References
[1] D. E. Blair and S. I. Goldberg, Topology of almost contact manifolds, J. Differ. Geom. Vol.1 (1967), 347-354.
[2] E. Calabi and B. Eckmann, A class of complex manifold which are not algebraic, Ann. of Math. vol.58 (1953), 494-500.
[3] Y. S. Cho, Hurwitz number of triple ramified covers, J. of Geom. and Phys. Vol.56, No.4 (2008), 542-555.
[4] Y. S. Cho, Quantum type cohomologies on contact manifolds, Inter. J. of Geom. Methods in Modern Phys. Vol.10, No.5 (2013).
[5] D. Janssens and J. Vanhecke, Almost contact structures and curvature tensors, Kodai Math. J. Vol.4 (1981), 1-27.
[6] J. Milnor, Morse Theory, Ann. of Math. studies Number 51, Princeton Univ. press, (1968).

The regularized mean curvature flow for invariant hypersurfaces in a Hilbert space

In this talk, I state some results for the regularized mean curvature flow starting from invariant hypersurfaces in a Hilbert space equipped with an isometric almost free Hilbert Lie group action whose orbits are minimal regularizable submanifolds. First we state the evolution equations for some geometric quantities along this flow. Some of the evolution equations are described by using the O’Neill fundamental tensor of the orbit map of the Hilbert Lie group action, where we note that the O’Neill fundamental tensor implies the obstruction for the integrability of the hor- izontal distribution of the orbit map. Next, by using the evolution equations, we derive some results for this flow. Furthermore, we derive some results for the mean curvature flow starting from compact Riemannian suborbifolds in the orbit space (which is a Riemannian orbifold) of the Hilbert Lie group action.
References
[1] R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255-306.
[2] G. Huisken, Contracting convex hypersurfaces in Riemannian manifolds by their mean cur- vature, Invent. math. 84 (1986) 463-480.
[3] E. Heintze, X. Liu and C. Olmos, Isoparametric submanifolds and a Chevalley-type restric- tion theorem, Integrable systems, geometry, and topology, 151-190, AMS/IP Stud. Adv. Math. 36, Amer. Math. Soc., Providence, RI, 2006.
[4] N. Koike, The mean curvature flow for invariant hypersurfaces in a Hilbert space with an almost free group action, arXiv:math.DG/1210.2539v2.
[5] B. O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966) 459- 469.
[6] C. L. Terng, Proper Fredholm submanifolds of Hilbert space, J. Differential Geom. 29 (1989) 9-47.
[7] C. L. Terng and G. Thorbergsson, Submanifold geometry in symmetric spaces, J. Differential Geom. 42 (1995) 665-718.

Calibrations and manifolds with special holonomy

Examples of n-dimensional Ricci flat manifolds are Riemannian manifolds whose holonomy groups Hol(g) are subgroups of SU(n), for n = 2m, and subgroups of the exceptional Lie group G2, for n = 7. We call them Calabi-Yau and G2 manifolds, respectively. They are also examples of manifolds with special holonomy. Calibrated submanifolds of Calabi-Yau and G2 manifolds are volume minimizing in their homology classes and their moduli spaces have many important applications in geometry, topology and physics. In particular, they are believed to play a crucial role in explaining the mysterious ”mirror symmetry” between pairs of Calabi-Yau and G2 manifolds. In this talk we give a report of recent research on the calibrations inside the manifolds with special holonomy.

On totally geodesic surfaces in symmetric spaces of type AI

We develop an approach to the classification of totally geodesic surfaces in Rie- mannian symmetric spaces of noncompact type. In this talk, we concentrate on the case of symmetric spaces of type AI, and mention that such surfaces correspond to certain nilpotent matrices. The classification results in the case of rank two and three will be described explicitly.
References
[1] B.-Y. Chen, T. Nagano, Totally geodesic submanifolds of symmetric spaces II, Duke Math. J. 45 (1978), 405–425.
[2] S. Klein, Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram, Geom. Dedicata 138 (2009), 25–50.
[3] S. Klein, Totally geodesic submanifolds in Riemannian symmetric spaces, in: Differential Geometry, 136–145, World Sci. Publ., Hackensack, NJ, 2009.
[4] K. Mashimo, Non-flat totally geodesic surfaces in symmetric spaces of classical type, preprint (2013).

Harmonic maps into Grassmannians

A harmonic map from a Riemannian manifold into a Grassmannian manifold is characterized by a vector bundle, a space of sections of the bundle and a Laplace operator.
This characterization can be considered a generalization of a theorem of Takahashi. We apply our main result which generalizes a theorem of do Carmo and Wallach to describe moduli spaces of special classes of harmonic maps from compact reductive Riemannian homogeneous spaces into Grassmannians.
As an application, we give an alternative proof of the theorem of Bando and Ohnita which states the rigidity of the minimal immersion of the complex projective line into complex projective spaces. Moreover, a similar method yields rigidity of holo- morphic isometric embeddings between complex projective spaces, which is part of Calabi’s result. Finally, we give a description of moduli spaces of holomorphic isometric embeddings of the projective line into quadrics.

The geometry on hyper-Kähler manifolds of type A∞

Hyper-K ̈ahler manifolds of type A∞ are noncompact complete Ricci-flat K ̈ahler manifolds whose homology groups are infinitely generated, which were constructed by Anderson, Kronheimer and LeBrun. They were also constructed by Goto as hyper-K ̈ahler quotients. In this talk I will talk about the asymptotic behavior of hyper-K ̈ahler metrics of these manifolds and about the holomorphic symplectic structures on them. Hyper-K ̈ahler manifolds of type A∞ have countably infinite parameters of deformations, and the parameters can be regarded as the cohomology classes of three real symplectic forms. I have described the volume growth of the hyper-K ̈ahler metrics using these parameters, and shown that there exists a complex manifold X of dimension 2 equipped with a family of Ricci-flat K ̈ahler metrics {gα}3<α<4, whose volume growth are given by rα, where r > 0 is a distance function from a point p ∈ X. Here, X is constructed as hyper-K ̈ahler manifolds of type A∞.
References
[1] T. Anderson, P. Kronheimer, C. LeBrun: Complete Ricci-flat K ̈ahler manifolds of infinite topological type,, Commun. Math. Phys., 125, (1989), 637-642.
[2] R. Goto: On hyper-Ka ̈hler manifolds of type A∞, Geom. Funct. Anal., 4, No. 4, (1994), 424-454.
[3] K. Hattori: The volume growth of hyper-K ̈ahler manifolds of type A∞, J. of Geometric Analysis, Vol. 21, No. 4, (2011), 920-949
[4] K. Hattori: The holomorphic symplectic structures on hyper-Ka ̈hler manifolds of type A∞, Advances in Geometry, to appear.

Elementary deformations and the hyperKaehler-quaternionic Kaehler correspondence

The hyperKaehler-quaternionic Kaehler correspondence constructs quaternionic Kaehler metrics from hyperKaehler metrics with a rotating circle symmetry. We discuss how this may be interpreted as a combination of the twist construction with the concept of elementary deformation, surveying results of our forthcoming paper. We outline how this leads to a uniqueness statement for the above correspondence and indicate how basic examples of c-map constructions may be realised in this contex.

Sequences of maximal antipodal sets of oriented real Grassmann manifolds

The author reduced the problem of classifying all maximal antipodal sets in the oriented real Grassmann manifold G ̃k(Rn) consisting of k-dimensional oriented subspaces in Rn to that of classifying all maximal antipodal subsets in the set Pk (n) consisting of subsets of cardinality k in {1, . . . , n} and classified all maximal antipodal subsets of Pk(n) for k ≤ 4 in the previous paper [2]. The notion of an antipodal set in a Riemannian symmetric space was introduced by Chen and Nagano [1]. The definition of antipodal subsets in Pk(n) comes from this, but it is completely combinatrial and defined by [2]. If k is more than 4, there may be so many maximal antipodal subsets of Pk(n) that it is difficult to classify all of them. In this talk, we construct some sequences of maximal antipodal subsets of Pk(n), which may be usefull for the classification of maximal antipodal subsets of Pk(n).
References
[1] B.-Y. Chen and T. Nagano, A Riemannian geometric invariant and its applications to a problem of Borel and Serre, Trans. Amer. Math. Soc. 308 (1988), 273–297.
[2] H. Tasaki, Antipodal sets in oriented real Grassmann manifolds, Internat. J. Math. 24 no.8 (2013), 1350061-1–28.

Cheeger constant, p-Laplacian, and Gromov-Hausdorff convergence

In this talk, we discuss a behavior of (λ1,p(M))1/p with respect to the Gromov- Hausdorff topology and the valuable p, where λ1,p(M) is the first positive eigenvalue of the p-Laplacian on a closed Riemannian manifold M. The behavior allows us to give a new isoperimetric inequality and a new Lichnerowicz-Obata type theorem.
References
[1] S. Honda, Ricci curvature and Lp-convergence, J. reine angew. Math. to appear.
[2] S. Honda, Cheeger constant, p-Laplacian, and Gromov-Hausdorff convergence,
arXiv:1310.0304, preprint.

The warped product approach to GMGHS spacetime

In the framework of Lorentzian multiply warped products we study the Gibbons- Maeda-Garfinkle-Horowitz-Strominger (GMGHS) spacetime and the nonsmooth geodesic motion near hypersurfaces in the interior of the event horizon. We also investigate the geodesics of the GMGHS spacetime with C0-warping functions.

Conformal transformation on complete product Riemannian manifold

T.Nagano proved that if the non-homothetic conformal transformation between complete Riemannian manifolds with parallel Ricci tensor is admitted, then the manifolds are irreducible and isometric to a sphere. From this result and other reasons, it is natural to ask for the problem that does there exist globally a non- homothetic conformal transformation between complete product Riemannian man- ifolds? In this talk, we introduce and consider about this question and related topics.

Lagrangian intersection theory and Hamiltonian volume minimizing problem

In 1990, Y.-G. Oh [1] posed variational problems of compact Lagrangian subman- ifolds in K ̈ahler manifolds under Hamiltonian deformations. Kleiner and Oh gave the first non-trivial example, namely, they showed that the standard RP n in CP n has the least volume under Hamiltonian deformations. To investigate Hamiltonian volume minimizing property of a Lagrangian submanifold, we can use the Arnold- Givental inequality in Lagrangian intersection theory and Crofton type formula in integral geometry.
Y.-G. Oh [2] also introduced the notion of tightness of a compact Lagrangian submanifold. A compact Lagrangian submanifold L in a homogeneous K ̈ahler man- ifold M is said to be globally tight if the cardinality of the intersection L ∩ gL is equal to the sum of Z2-Betti numbers of L whenever L and gL intersect transversely for an isometry g of M. In this talk, we first discuss antipodal sets and the global tightness of real forms in Hermitian symmetric spaces and complex flag manifolds. As applications, we calculate Lagrangian Floer homology and obtain a general- ization of the Arnold-Givental inequality. Moreover, we apply it to Hamiltonian volume minimizing problems.
References
[1] Y.-G. Oh, Second variation and stabilities of minimal Lagrangian submanifolds in K ̈ahler manifolds, Invent. Math. 101 (1990), 501–519.
[2] Y.-G. Oh, Tight Lagrangian submanifolds in CPn, Math. Z. 207 (1991), 409–416.
[3] H. Iriyeh, T. Sakai and H. Tasaki, Lagrangian Floer homology of a pair of real forms in
Hermitian symmetric spaces of compact type, J. Math. Soc. Japan 65 (2013), 1135–1151.
[4] H. Iriyeh, T. Sakai and H. Tasaki, On the structure of the intersection of real flag manifolds
in a complex flag manifold, to appear in Adv. Stud. Pure Math.

Canonical connection on contact manifolds

we explain the analysis of the following system of elliptic equation ∂πw = 0, d(w∗λ ◦ j) = 0
associated for each given contact triad (Q, λ, J) on a contact manifold (Q, ξ). We directly work with this equation on the contact manifold without involving the symplectization process. We explain the basic analytical ingredients towards the construction of moduli space of solutions, which we call contact instantons. We will indicate how one could define contact homology type invariants using such a moduli space, which is still in progress. This is partially based on the joint work with Rui Wang.

Geometry of Lagrangian submanifolds related to isoparametric hypersurfaces

In this talk we shall provide a survey of my recent works and their environs on dif- ferential geometry of Lagrangian submanifolds in specific K ̈ahler manifolds, such as complex projective spaces, complex space forms, Hermitian symmetric spaces and so on. We shall emphasis on the relationship between certain minimal La- grangian submanifold in complex hyperquadrics and isoparametric hypersurfaces in spheres. We shall discuss their properties and related problems such as classifi- cation, Hamiltonian stability and intersection of the Gauss images of isoparametric hypersurfaces. This talk is mainly based on my joint work with Hui Ma (Tsinghua University, Beijing).

2014-08-12

The recent progress in the isoparametric functions and isoparametric hypersurfacesThis talk will give a survey in the recent progress in the isoparametric functions and isoparametric hypersurfaces, especially on two directions.
(1)isoparametric functions on exotic spheres. The existences and non-existences will be considered.
(2)Yau conjecture on the first eigenvalues of the minimal hypersurfaces in the unit spheres. The history of progress of Yau conjecture on minimal isoparametric hypersurfaces.

Cho operators on real hypersurfaces in complex projective space

Let M be a real hypersurface in complex projective space. On M we have the Levi-Civita connection and for any nonzero constant k the corresponding generalized Tanaka-Webster connection. For such a k and any tangent vector field X to M we can define from both connections the k-th Cho Operator F(k). We study X commutativity properties of these operators with the shape operator and the struc- ture Jacobi operator on M obtaining some characterizations of either type (A) real hypersurfaces or ruled real hypersurfaces.

Construction of coassociative submanifolds

The notion of coassociative submanifolds is defined as the special class of the min- imal submanifolds in G2-manifolds. In this talk, we introduce the method of [1] to construct coassociative submanifolds by using the symmetry of the Lie group action. As an application, we give explicit examples in the 7-dimensional Euclidean space and in the anti-self-dual bundle over the 4-sphere.
References
[1] W. Y. Hsiang and H. B. Lawson, Minimal Submanifolds of Low Cohomogeneity, J. Differ- ential Geom. 5, (1971), 1-38.
[2] K. Kawai, Construction of Coassociative submanifolds in R7 and Λ2−S4 with symmetries, preprint, math.DG/1305.2786.

Geominimal surface area and its extension

We present some Lp affine isoperimetric inequalities for Lp geominimal surface area. In particular, we obtain an analogue of Blaschke-Santal ́o inequality and a cyclic inequality for Lp geominimal surface areas. We give an integral formulas of Lp geominimal surface area by the p-Petty body. Furthermore, we introduce the concept of Lp mixed geominimal surface area which is a nature extension of Lp geominimal surface area. We also extend Lutwak’s results for Lp mixed geominimal surface area.
References
[1] A.D. Aleksandrov, On the theory of mixed volumes. i. Extension of certain concepts in the theory of convex bodies, Mat. Sb. (N. S.) 2 (1937), 947-972. [Russian].
[2] S. Alesker, Continuous rotation invariant valuations on convex sets, Ann. of Math. 149 (1999), 977-1005.
[3] S. Alesker, Description of translation invariant valuations on convex sets with a solution of P. McMullen’s conjecture, Geom. Funct. Anal. 11 (2001), 244-272.
[4] W. Blaschke, Vorlesungen ber Differentialgeometrie II, Affine Differentialgeometrie, Springer-Verlag, Berlin, 1923.
[5] J. Bourgain and V.D. Milman, New volume ratio properties for convex symmetric bodies in Rb, Invent. Math. 88 (1987), 319-340.
[6] W. Fenchel and B. Jessen, Mengenfunktionen und konvexe k ̈ooper, Danske Vid. Selskab. Mat.-fys. Medd. 16 (1938), 1-31.
[7] W.J. Firey, p-means of convex bodies, Math. Scand. 10 (1962), 17-24.
[8] P.M. Gruber, Aspects of approximation of convex bodies, Handbook of Convex Geometry, vol.A, 321-345, North Holland, 1993.
[9] P.M. Gruber, Convex and discrete geometry, Springer-Verlag, Berlin Heidelberg, 2007.
[10] G.H. Hardy, J.E. Littlewood and G. P ́olya, Inequalities, Cambridge Univ. Press, London, 1934.
[11] J. Jenkinson and E. Werner, Relative entropies for convex bodies, to appear in Trans. Amer. Math. Soc.
[12] G. Kuperberg, From the Mahler conjecture to Gauss linking integrals, Geom. Funct. Anal. 18 (2008), 870-892.

Hamiltoian minimality of normal bundles over the isoparametric submanifolds

A Lagrangian submanifold in a K ̈ahler manifold is called Hamiltonian minimal (shortly, H-minimal) if it has extremal volume under Hamiltonian deformations. In this talk, we give new families of non-compact H-minimal Lagrangian submanifolds in the complex Euclidean space Cn. Let G be a compact semi-simple Lie group, and g be a Lie algebra of G. We show that any normal bundle of a principal orbit in g of the adjoint representation of G is an H-minimal Lagrangian submanifold in the tangent space Tg which is naturally identified with Cn. Moreover, we characterize these orbits by this property among the class of full, irreducible isoparametric submanifolds in the Euclidean space.

Real hypersurfaces in complex two-plane Grassmannians with recurrent structure Jacobi operator

In this talk, we introduce a new notion of recurrent structure Jacobi operator, that is, (∇XRξ)Y = ω(X)RξY for any tangent vector fields X and Y on a real hyper- surface M in a complex two-plane Grassmannian, where Rξ denotes the structure Jacobi operator and ω a certain 1-form on M. Next, we show that there does not exist any Hopf hypersurface M in a complex two-plane Grassmannian with recurrent structure Jacobi operator.
References
[1] J. Berndt and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians, Monat- shefte fu ̈r Math. 127 (1999), 1–14.
[2] I. Jeong, J. D. P ́erez and Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with parallel structure Jacobi operator, Acta Math. Hungar. 122 (2009), no. 1-2, 173-186.
[3] I. Jeong, Machado, Carlos J. D. P ́erez and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with D⊥-parallel structure Jacobi operator, Internat. J. Math. 22 (2011), no. 5, 655-673.
[4] I. Jeong, H. Lee and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians whose structure Jacobi operator is of Codazzi type, Acta Math. Hungar. 125 (2009), no. 1-2, 141-160.
[5] J. D. P ́erez and F. G. Santos, Real hypersurfaces in complex projective space with recurrent structure Jacobi operator, Diffential geometry and its Applications 26 (2008), 218–223.

Some characterizations of real hypersurfaces in complex hyperbolic two-plane Grassmannians

A main objective in submanifold geometry is the classification of homogeneous hypersurfaces. Homogeneous hypersurfaces arise as principal orbits of cohomo- geneity one actions, and so their classification is equivalent to the classification of cohomogeneity one actions up to orbit equivalence. Actually, the classification of cohomogeneity one actions in irreducible simply connected Riemannian symmetric spaces of noncompact type was obtained by J. Berndt and Y.J. Suh (for complex hyperbolic two-plane Grassmannain SU2,m/S(U2·Um), [1]).
From this classification, in [6] Suh classified real hypersurfaces with isometric Reeb flow in SU2,m/S(U2·Um), m ≥ 2. Each can be described as a tube over a totally geodesic SU2,m−1/S(U2·Um−1) in SU2,m/S(U2·Um) or a horosphere whose center at infinity is singular. By using this result, we want to give another characterization for these model spaces by the Reeb invariant shape operator, that is, LξA = 0.
References
[1] J. Berndt and Y. J. Suh, Hypersurfaces in noncompact complex Grassmannians of rank two, International J. Math. 23 (2012), 1250103, 35pp.
[2] J. Berndt and Y. J. Suh, Contact hypersurfaces in K ̈aehler manifolds, Proc. Amer. Math. Soc. (2014, in press).
[3] J. Berndt, H. Lee and Y. J. Suh, Contact hypersurfaces in noncompact complex Grassman- nians of rank two, International J. Math. 24 (2013), 13500, 11pp.
[4] S. Dragomir and D. Perrone, Harmonic vector fields: Variational principles and differential geometry, Elsevier, 2011.
[5] Y.J. Suh, Real hypersurfaces in complex hyperbolic two-plane Grassmannians related to the Reeb vector field, Adv. in Appl. Math. 51 (2014, in press).
[6] Y.J. Suh, Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grass- mannians, Adv. in Appl. Math. 50 (2013), 645–659.

Parallelism on Jacobi operator for Hopf hypersurfaces in complex two-plane Grassmannians

In relation to the generalized Tanaka-Webster connection, we introduce a new no- tion of parallel Jacobi operator for real hypersurfaces in complex two-plane Grass- mannians G2(Cm+2). Next we prove some new results for real hypersurfaces in G2(Cm+2) with parallel structure and normal Jacobi operators in the generalized Tanaka-Webster connection.
References
[1] J. Berndt, Riemannian geometry of complex two-plane Grassmannian, Rend. Sem. Mat. Univ. Politec. Torino 55 (1997), 19–83.
[2] I. Jeong, H. J. Kim and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with parallel normal Jacobi operator, Publ. Math. Debrecen 76 (2010), 203–218.
[3] H. Lee, J. D. P ́erez and Y.J. Suh, On the structure Jacobi operator of a real hypersurface in complex projective space, Monatsh. Math. 158 (2009), no. 2, 187–194.
[4] E. Pak and Y. J. Suh, Hopf hypersurfaces in complex two-plane Grassmannians with general- ized Tanaka-Webster D⊥-parallel structure Jacobi operator, Cent. Eur. J. Math., 12 (2014), no. 5, (in press).

Real hypersurfaces in complex two-plane Grassmannians with commuting Jacobi operators

In this talk, we introduce a new notion of recurrent structure Jacobi operator, that is, (∇XRξ)Y = ω(X)RξY for any tangent vector fields X and Y on a real hyper- surface M in a complex two-plane Grassmannian, where Rξ denotes the structure Jacobi operator and ω a certain 1-form on M. Next, we show that there does not exist any Hopf hypersurface M in a complex two-plane Grassmannian with recurrent structure Jacobi operator.
References
[1] J. Berndt and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians, Monat- shefte fu ̈r Math. 127 (1999), 1–14.
[2] I. Jeong, J. D. P ́erez and Y.J. Suh, Real hypersurfaces in complex two-plane Grassmannians with parallel structure Jacobi operator, Acta Math. Hungar. 122 (2009), no. 1-2, 173-186.
[3] I. Jeong, Machado, Carlos J. D. P ́erez and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with D⊥-parallel structure Jacobi operator, Internat. J. Math. 22 (2011), no. 5, 655-673.
[4] I. Jeong, H. Lee and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians whose structure Jacobi operator is of Codazzi type, Acta Math. Hungar. 125 (2009), no. 1-2, 141-160.
[5] J. D. P ́erez and F. G. Santos, Real hypersurfaces in complex projective space with recurrent structure Jacobi operator, Diffential geometry and its Applications 26 (2008), 218–223.

Heat content asymptotics on a compact Riemannian manifold with boundary

We review the heat content asymptotics on a compact Riemannian manifold with boundary and with specific heat and initial temperature distributions. Some com- putation of first a few terms in the asymptotics series are shown given the existence of a complete asymptotics series. This is a joint work with Michiel van den Berg and Peter Gilkey.
References
[1] M. van den Berg and E. B. Davies, Heat flow out of regions in Rm, Math. Z. 202. (1989), 463–482.
[2] M. van den Berg and P. Gilkey, Heat content asymptotics of a Riemannian manifold with boundary, J. Funct. Anal. 120. (1994), 48–71.
[3] M. van den Berg and P. Gilkey, Heat invariants for odd dimensional hemispheres, Proc. Edinb. Math. Soc. 126A. (1996), 187–193.
[4] M. van den Berg and P. Gilkey, Heat equation with inhomogeneous Dirichlet conditions, Comm. Analysis and Geometry 7. (1999), 279–294.
[5] M. van den Berg and P. Gilkey, Heat content asymptotics with singular data, J. Phys. A: Math. Theor. 45. (2012), 374027
[6] M. van den Berg, P. Gilkey, A. Grigor’yan and K. Kirsten, Hardy inequality and heat semi- group estimates for Riemannian manifolds with singular data, Comm. Part. Diff. Eq. 37. (2012), 885–900.
[7] M. van den Berg, P. Gilkey and K. Kirsten Growth of heat trace and heat content asymptotic coefficients, J. Funct. Anal. 261. (2011), 2293–2322.
[8] M. van den Berg, P. Gilkey, K. Kirsten and V. A. Kozlov, Heat Content Asymptotics for Riemannian manifolds with Zaremba boundary conditions, Potential Analysis 26. (2007), 225–254.
[9] M. van den Berg, P. Gilkey and R. Seeley, Heat Content Asymptotics with singular initial temperature distributions, J. Funct. Analysis 254. (2008), 3093–3122.
[10] M. V. Berry and C. J. Howls, High orders of the Weyl expansion for quantum billiards: resurgence of periodic orbits and the Stokes phenomenon, Proc. R. Soc. Lond. A 447. (1994), 527–555.
[11] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Clarendon Press, Oxford (2000).

Totally geodesic submanifolds of Riemannian symmetric spaces

A submanifold Σ of a Riemannian manifold M is said to be totally geodesic if every geodesic in Σ is also a geodesic in M. The existence and classification of totally geodesic submanifolds are two fundamental problems in submanifold geometry. The totally geodesic submanifolds in Riemannian symmetric spaces of rank one were classified by Wolf ([8]). Klein obtained in a series of papers ([3], [4], [5], [6]) the classification in irreducible Riemannian symmetric spaces of rank two. No full classifications are known for higher rank. A rather well-known result states that an irreducible Riemannian symmetric space which admits a totally geodesic hypersurface must be a space of constant curvature. The first proof of this fact was given by Iwahori ([2]). Onishchik introduced in [7] the index of a Riemannian symmetric space M as the minimal codimension of a totally geodesic submanifold of M. He then gave an alternative proof for Iwahori’s result and also classified the irreducible Riemannian symmetric spaces with index 2. In the talk I will present recent results obtained in joint work with Carlos Olmos ([1]). The first result states that the index of an irreducible Riemannian symmetric space is bounded from below by the rank of the symmetric space. The second result is the classification of all irreducible Riemannian symmetric spaces whose index is less or equal than three.
References
[1] J. Berndt and C. Olmos, On the index of symmetric spaces, preprint arXiv:1401.3585.
[2] N. Iwahori, On discrete reflection groups on symmetric Riemannian manifolds, in: Proc. US-Japan Seminar in Differential Geometry (Kyoto 1965), Nippon Hyoronsha, Tokyo (1966), 57–62.
[3] S. Klein, Totally geodesic submanifolds of the complex quadric, Differential Geom. Appl. 26 (2008), 79–96.
[4] S. Klein, Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram, Geom. Dedicata 138 (2009), 25–50.
[5] S. Klein, Totally geodesic submanifolds of the complex and the quaternionic 2-Grassmannians, Trans. Amer. Math. Soc. 361 (2010), 4927–4967.
[6] S. Klein, Totally geodesic submanifolds of the exceptional Riemannian symmetric spaces of rank 2, Osaka J. Math. 47 (2010), 1077–1157.
[7] A.L. Oniwik, O vpolne geodeziqeskih podmnogoobrazi