Submanifold Geometry, Lie Group Action and Its Applications to Theoretical Physics 2024

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Weinjiao Yan (Beijing Normal University, China): 2-talks, in-person
Joseph Ansel Hoisington (Colby College, USA): in-person
Jong Taek Cho (Chonnam National University, Korea): online
Giuseppe Pipoli (Univesità degli Studi dell'Aquila, Italy): online
Kurando Baba (Tokyo University of Science)
Naotoshi Fujihara (Tokyo University of Science)
Tomoki Fujii (Tokyo University of Science)
Kota Hattori (Keio University)
Yasushi Homma (Waseda University)
Toru Kajigaya (Tokyo University of Science)
Keita Kunikawa (Tokushima University)
Masahiro Morimoto (Tokyo Metropolitan University)
Toshihiro Shoda (Kansai University)
Daisuke Tarama (Ritsumeikan University)

Kurando Baba (Tokyo University of Science)
"Toward a construction of special Lagrangian submanifolds of the Atiyah-Hitchin manifold"
Harvey and Lawson (1982) introduced the notion of calibrated submanifolds in a general Riemannian manifold. It is shown that these submanifolds are homologically volume-minimizing. As a concrete example of calibrated submanifolds, they provided special Lagrangian submanifolds in a Calabi-Yau manifold. Special Lagrangian submanifolds have been extensively studied not only in mathematics but also in mathematical physics. The construction of special Lagrangian submanifolds gives a fundamental problem. One approach to this is a construction method applying the moment map, and previous studies were given by Noda (2008), Hashimoto-Sakai (2012), and Koike (2019) etc.. In this talk, we will explain some results toward a construction of special Lagrangian submanifolds in the Atiyah-Hitchin manifold, known as a 4-dimensional hyperk\"ahler manifold, by using the moment map. This talk is based on a joint work with Masato Arai (Yamagata University).

Jong Taek Cho (Chonnam National University, Korea): online
"Contact structures, CR-structures and hypersurface geometry"
Given a contact one-form, we have the two fundamental associated structures, that is, a Riemannian metric and an almost CR-structure. In this talk, we first review the previous works which imply that (Cartan’s) local symmetry is too strong to impose on contact manifolds. Next, we introduce a notion of pseudo-Hermitian symmetry from the viewpoint of almost CR-structures. Then we give a complete classification of non-Sasakian contact manifolds with the pseudo-Hermitian symmetry. Indeed, they are realized as real hypersurfaces of the complex quadric, the non-compact dual space and the complex Euclidean space. Moreover, we give a classification of pseudo-Hermitian symmetric CR-space forms of dimension greater than 3 and the dimension 3.

Naotoshi Fujihara (Tokyo University of Science)
"Mean Curvature Flows of Graphs in Warped Products"
In a warped product with an open interval as the base and a closed Riemannian manifold as the fiber, we define graphical hypersurfaces and study the mean curvature flow of these hypersurfaces. In this talk, I will discuss the graph-preserving property of the mean curvature flow and the long-time existence of graphical mean curvature flows. Additionally, I will explain certain properties specific to the curve shortening flow, and the invariance of the mean curvature flow under certain Lie group actions.

Tomoki Fujii (Tokyo University of Science)
"Graphical solitons invariant under hyperpolar actions"
In this talk, we discuss graphical solitons for the mean curvature flow. First, we show the classification of shapes of translators given by a graph of a function on the sphere which is constant along each leaf of isoparametric foliation. Next, we consider solitons given by a graph of a function on the symmetric space of compact type. We show the classification of shapes of such translators in the case that the rank of the symmetric space is equal to one and the function is invariant under the isotropy action. Also, in the case that the symmetric space is of higher rank, we introduce the graphical translators invariant under the Hermann action of cohomogeneity two. Moreover, we state the recent works on the isotropy invariant graphical rotating solitons with codimension two over the symmetric space.

Kota Hattori (Keio University)
"The energy of smooth maps and Gibbons-Hawking metrics"
A critical point of the Dirichlet energy for smooth maps is called a harmonic map. In general, harmonic maps do not minimize the energy. In this talk, I will explain the notion of calibrated geometry for smooth maps and show examples given as the hyper-Kahler moment maps associate with the Gibbons-Hawking metrics.

Joseph Ansel Hoisington (Colby College, USA): in-person
"Energy-minimizing maps of real and complex projective spaces"
We show that the infimum of the energy in a homotopy class of maps from complex projective space to a Riemannian manifold is proportional to the infimal area in the homotopy class of maps of the 2-sphere representing the induced homomorphism on the second homotopy group. We then establish a two-sided estimate for the infimum of the energy in a homotopy class of maps from real projective space to a Riemannian manifold, as well as a related estimate for a wider class of energy functionals. Together, these results suggest that the problem of determining the infimum of the energy in a homotopy class, solved for complex projective space by our first result, may be more complicated for real projective space.

Yasushi Homma (Waseda University)
"Weitzenb\"ock formulas and their applications on quaternionic K\"ahler manifolds"
Weitzenb\"ock formulas are useful in various aspects of differential geometry. On a quaternion K\"ahler manifolds, their differential geometry and representation theory of $Sp(1)Sp(n)$ are closely intertwined, and various geometric results can be obtained as applications of Weitzenb\"ock formulas. In this talk, I will explain how to obtain those formulas and discuss various applications. In particular, I would like to explain recent results obtained in collaboration with U. Semmelmann.

Toru Kajigaya (Tokyo University of Science)
"Index estimate by first Betti number of minimal hypersurface in compact symmetric space: Part II"
Following Kunikawa's talk, I will talk about outline of the proof of the result in our joint work. The proof of the index estimate is inspired by the methods of Savo and Ambrozio-Carlotto-Sharp. Their approach is based on a kind of "averaging method" using isometric immersion into the Euclidean space. Our key observation in our proof is that this method can be naturally extended by considering isometric immersion into a compact semi-simple Riemannian symmetric space. I will explain in detail how this method works.

Keita Kunikawa (Tokushima University)
"Index estimate by first Betti number of minimal hypersurface in compact symmetric space: Part I"
We show that the Morse index of unstable closed minimal hypersurface $\Sigma$ in a compact semi-simple Riemannian symmetric space $M=G/K$ is bounded from below by constant times the first Betti number of $\Sigma$, where the constant depends only on $M$. This provides a partial answer to the conjecture of Schoen, Marques and Neves concerning a similar bound holds for any closed minimal hypersurfacde in a closed Riemannian manifold of positive Ricci curvature. This talk is based on joint research with Toru Kajigaya (Tokyo University of Science), and is divided into two parts: Part I, presented by Kunikawa, will provide an overview of previous studies and related works on Morse index estimates for minimal hypersurfaces, while Part II, presented by Kajigaya, will explain the details of our proof.

Masahiro Morimoto (Tokyo Metropolitan University)
"The parallel transport map over affine symmetric space"
C.-L. Terng and G. Thorbergsson investigated a natural Riemannian submersion from an infinite dimensional Hilbert space onto a compact Riemannian symmetric space, which is called the parallel transport map. Afterward, N. Koike extended their theory to the case of Riemannian symmetric space of non-compact type. In this talk, I will talk about my research on the generalization of those theories in the framework of affine differential geometry.

Giuseppe Pipoli (Univesità degli Studi dell'Aquila, Italy): online
"Vanishing theorems for minimal stable hypersurfaces"
We will discuss some topological obstructions to the existence of stable minimal hypersurfaces. In particular, we will show the non-existence of nontrivial harmonic p-forms and nontrivial harmonic spinors on stable minimal hypersurfaces under suitable curvature assumptions of the ambient manifold. This talk is based on a joint work with Francesco Bei (Sapienza, Università di Roma).

Toshihiro Shoda (Kansai University)
"On a set of index one holomorphic maps of compact Riemann surfaces onto the two-sphere"
The index of a holomorphic map from a compact Riemann surface onto the two-sphere was introduced by Montiel-Ros in order to describe the Morse index of a minimal surface. In particular, it is important to study an index one holomorphic map from the viewpoint of stability. It is known that there exist many genus three examples of triply periodic minimal surfaces whose Gauss maps have indices one. On the other hand, the Bolza surface, which is a compact Riemann surface of genus two, has index one holomorphic map. In this talk, we shall report that there exists a set of index one holomorphic maps which contains the former holomorphic maps and which approaches the latter one.

Daisuke Tarama (Ritsumeikan University)
"Left-invariant geodesic flows of Lie groups and free rigid body dynamics"
This talk deals with the Hamiltonian system associated with the geodesic flow of a Lie group equipped with a left-invariant metric. If the Lie group is SO(3), the system describes the rotational motion of a free rigid body, which is a typical example of integrable systems in analytical mechanics. As a generalization of this system, Mishchenko and Fomenko introduced a class of left-invariant metrics on a semi-simple Lie group whose geodesic flows are completely integrable. The main part of the talk is concerned with the dynamical properties of (relative) equilibrium points for the geodesic flows of the semi-simple Lie groups with respect to the Mishchenko-Fomenko metric. Particularly, a characterization of Williamson types of isolated equilibrium points on generic coadjoint orbits is given in terms of root systems. In the rest of the talk, further recent developments are mentioned. The talk includes a collaboration with Tudor Ratiu (Shanghai Jiao Tong Univ.).

Weinjiao Yan (Beijing Normal University, China): 2 talks, in-person
"Orthogonal almost complex structure and its Nijenhuis tensor"
We demonstrate that on an almost Hermitian manifold (M^2n, J, ds^2), a 2-form ϕ = S*Φ, the pulling back of the K¨ahler form Φ on the twistor bundle over M^2n , is non-degenerate if the squared norm |N|^2 of the Nijenhuis tensor is less than 64/5 when n ≥ 3 or less than 16 when n = 2. As one of consequences, there exists no orthogonal almost complex structure on the standard sphere S^6 with |N|^2 < 64/5 everywhere. This talk is based on joint work with Professor Zizhou Tang.
"Isoparametric foliation and complex structures"
We construct complex structures on a series of isoparametric hypersurfaces and focal submanifolds in the unit sphere. As a special case, we obtain a complex structure on the product of S^6 and N^8 (a closed manifold). This talk is based on joint works with Professor Zizhou Tang and Professor Chao Qian.