Workshop on Submanifolds and Harmonic maps

Department of mathematics, Tongji University

4.18-4.19, 2015

1. Lectures on Mean curvature flows

Entao Zhao (Zhejiang University)

Abstract: In first two lectures, I will firstly introduce some basics of the mean curvature flow, including the definition, the short-time existence, basic properties, and evolution equations of geometric quantities, etc., then I will talk about the smooth convergence of the mean curvature flow of convex hypersurfaces in the Euclidean space and briefly introduce the singularity analysis of mean curvature flow on hypersurfaces. In the third lecture, I will talk about some of our recent works on the mean curvature flow of higher codimensions.

Reference:

[1] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266.

[2] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom., 31(1990), 285-299.

[3] G. Huisken and C. Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Differ. Equ., 8(1999), 1-14.

[4] K. F. Liu, H. W. Xu and E. T. Zhao, Mean curvature flow of higher codimension in Riemannian manifolds, arXiv:1204.0107.

[5] X. P. Zhu, Lectures on mean curvature flows, Studies in Advanced Mathematics, Vol. 32, International Press, 2002.

[6] C. Mantegazza, Lecture notes on mean curvature flow, Progress in Mathematics, Vol. 290, Birkhauser, 2010.

[7] K. Ecker, Regularity theory for mean curvature flow, Progress in Nonlinear Differential Equations and their Applications, Vol. 57, Birkhauser, 2004.

1. Minimal two-spheres with constant curvature in Grassmannians

Xiaowei Xu (University of Science and Technology of China)

Abstract: In this talk, I will introduce that how to classify the homogeneous minimal two-spheres in complex Grassmann manifold G(2,n). Some other related results are introduced.

Reference:

[1] Peng, Chiakuei; Xu, Xiaowei Classification of minimal homogeneous two-spheres in the complex Grassmann manifold G(2,n). J. Math. Pures Appl. (9) 103 (2015), no. 2, 374–399.

[2] Peng, Chiakuei; Xu, Xiaowei Minimal two-spheres with constant curvature in the complex Grassmannians. Israel J. Math. 202 (2014), no. 1, 1–20.

1. Submanifolds with constant Jordan angles

Ling Yang (Fudan University)

Abstract: To study the Lawson-Osserman's counterexample to the Bernstein problem for minimal submanifolds of higher codimension, a new geometric concept, submanifolds in Euclidean space with constant Jordan angles(CJA), is introduced in this paper, which is a generalization of constant angle curves and surfaces. By exploring the second fundamental form of submanifolds with CJA, we can characterize the Lawson-Osserman's cone from the viewpoint of Jordan angles.

Reference:

[1] J. Jost, Y. L. Xin, Ling Yang Submanifolds with constant Jordan angles arXiv:1502.02797

1. Isoparametric functions on exotic spheres.

Chao Qian (University of Chinese Academy of Sciences)

Abstract: We will discuss isoparametric functions on general Riemannian manifolds. In particular, the existence and non-existence results on exotic spheres are considered.

Reference:

[1] Qian C, Tang Z. Isoparametric functions on exotic spheres[J]. Advances in Mathematics, 2015, 272: 611-629.

1. Lectures on harmonic tori

Erxiao Wang

Abstract: We will give an introduction to some integrable system approaches to the construction and classification of harmonic tori "of finite type" in symmetric spaces.

Reference:

[1] Hitchin, Segal, Ward, Integrable Systems: Twistors, Loop groups, and Riemann Surfaces.

[2] F. E. Burstall, D. Ferus, F. Pedit and U. Pinkall, Harmonic Tori in Symmetric Spaces and Commuting Hamiltonian Systems on Loop Algebras, Annals of Mathematics, Second Series, Vol. 138, No. 1 (Jul., 1993), pp.173-212.