Special Session 91: Geometric evolution problems

Analysis of Critical Points of Conformally Invariant Curvature Energies in 4d

Yann Bernard Monash University Australia Co-Author(s):    Tian Lan, Dorian Martino, Tristan Riviere

Abstract: We consider the Euler--Lagrange equations satisfied by the critical points of a large class of conformally invariant extrinsic energies for 4-manifolds immersed into Euclidean space. Using invariances and Noether`s theorem, we convert the Euler--Lagrange equation into a system of equations with analytically favourable structures, and we develop small-energy estimates. This generalises to the four-dimensional setting ideas originally developed for the Willmore energy in two dimensions.

Degenerate and Singular Fractional Equations and Optimally Embedded Submanifolds

Simon SB Blatt University Salzburg Austria Co-Author(s):

Abstract: Over the past three decades, a variety of energy functionals have been developed and analyzed to address the problem of finding optimal embeddings of curves, links, and higher-dimensional submanifolds into Euclidean space. The associated Euler-Lagrange equations give rise to a rich class of geometric non-local differential equations, encompassing degenerate elliptic, singular, critical, and subcritical types. In this talk, we will provide a concise overview of the state of the art in this field, highlighting key results, challenges, and open questions

Classifying ancient solutions in mean curvature flow

Beomjun Choi KAIST Korea Co-Author(s):

Abstract: Ancient solutions play a central role in mean curvature flow, since they often arise as blow-up models near singularities. This leads to a natural program: by classifying ancient solutions, one hopes to organize the possible local pictures of singularity formation. In this talk, I will give an introduction to this viewpoint and describe several recent classification results, focusing in particular on ancient ovals and related rigidity phenomena in higher dimensions. I will also explain how such results fit into the broader effort to understand singularities through canonical models.

Integral curvature bounds and Perelman`s bounded diameter conjecture for Type I Ricci flows

Panagiotis Gianniotis National and Kapodistrian University of Athens Greece Co-Author(s):    Konstantinos Leskas

Abstract: Given a smooth Ricci flow that becomes singular in finite time, a $k$-neck regions is a region of the manifold in which the flow is almost self-similar and almost splits $k$ Euclidean factors, down to arbitrarily small scales. Neck regions are characterized by a set of centres which can be thought of as a discrete approximation of high curvature regions. In this talk we will describe how, under a Type I curvature bound, we can effectively control the $k$-dimensional size of the set of centres, by carefully analyzing the behaviour of almost splitting maps at small scales. As an application, we will see how such a result implies Perelman`s bounded diameter conjecture for a 3d Ricci flow exhibiting Type I singularities. In higher dimensions, it can be used to obtain the optimal $L^1$ curvature bound in higher dimensions, in joint work with Konstantinos Leskas.

Rigidity of minimal and CMC hypersurfaces.

Han Hong Beijing Jiaotong University Peoples Rep of China Co-Author(s):

Abstract: This talk will present a broad overview of rigidity results for stable and finite-index minimal (or constant mean curvature) surfaces. We will survey key milestones in the field, including the stable Bernstein problem for minimal hypersurfaces, its capillary surface analogue, do Carmo`s problem for non-minimal CMC hypersurfaces, and rigidity theorems for stable minimal hypersurfaces in general Riemannian manifolds. The presentation will focus on the statements of these results and their geometric significance, without delving into the technical details of the proofs.

Capillary Christoffel-Minkowski problem

Yingxiang Hu Beihang University Peoples Rep of China Co-Author(s):    Mohammad N. Ivaki (TU Wien) and Julian Scheuer (Goethe University Frankfurt)

Abstract: The result of Guan and Ma (Invent. Math. 151 (2003)) states that if $\phi^{-1/k} : \mathbb{S}^n \to (0,\infty)$ is spherically convex, then $\phi$ arises as the $\sigma_k$ curvature (the $k$-th elementary symmetric function of the principal radii of curvature) of a strictly convex hypersurface. We establish an analogous result in the capillary setting in the half-space for $\theta\in (0,\pi/2)$: if $\phi^{-1/k} : \mathcal{C}_{\theta} \to (0,\infty)$ is a capillary function and spherically convex, then $\phi$ is the $\sigma_k$ curvature of a strictly convex capillary hypersurface. This is based on joint works with Mohammad N. Ivaki (TU Wien) and Julian Scheuer (Goethe University Frankfurt).

The $p$-elastic flow of inextensible planar curves

Chun-Chi Lin National Taiwan Normal University Taiwan Co-Author(s):

Abstract: In this talk, I will discuss the existence of solutions to the negative $L^2$-gradient flow of the $p$-elastic energy for the class of inextensible planar closed or open curves. For open curves, the boundary conditions correspond to either hinged ends (i.e., zero curvature at their boundaries) or clamped ends (i.e., fixed contact angles at their boundaries). We show the existence of weak solutions to the negative $L^2$-gradient flow for $p\in(1,\infty)$.

The conical K\"ahler-Ricci flow and its limit behavior

Jiawei Liu Nanjing University of Science and Technology Peoples Rep of China Co-Author(s):    Xi Zhang

Abstract: In this talk, I will recall some results on conical K\"ahler-Ricci flow, and then talk about its limit behavior as the cone angle tends to 0. More precisely, as the cone angle tends to zero, the conical K\"ahler-Ricci flow converges to a unique K\"ahler-Ricci flow, which is smooth outside the divisor and admits cusp singularity along the divisor.

Stabilization technique applied on curve shortening flow in R^2 and R^3

Hayk Mikayelyan University of Nottingham Ningbo China Peoples Rep of China Co-Author(s):

Abstract: irst we apply the stabilization technique, developed by T. Zelenyak in 1960s for parabolic equations, on the curve shortening flow in $\mathbb{R}^2$, and derive a new monotonicity formula with logarithmic terms. Then we use this idea and derive several new monotonicity formulas for the CSF in $\mathbb{R}^3$. All of them share one main feature: the dependence of the ``energy`` term on the angle between the position vector and the plane orthogonal to the tangent vector.

Direction energy method for non-compact mean curvature flows

Tatsuya Miura Kyoto Univeristy Japan Co-Author(s):    Tatsuya Miura

Abstract: We introduce the direction energy for oriented immersions into Euclidean space in arbitrary dimensions and codimensions. Our main result establishes an exact dissipation identity for the direction energy along complete, proper, oriented mean curvature flows under a mild local-volume bound. We also discuss several geometric applications. This talk is based on joint work with Fabian Rupp (University of Vienna).

Integral Gauss formula and the Poisson equation for the G$_2$-Laplacian

Artem Pulemotov The University of Queensland Australia Co-Author(s):    Timothy Buttsworth, Stepan Hudecek, Artem Pulemotov

Abstract: The study of special holonomy involves a nonlinear second-order operator on differential forms called the G$_2$-Laplacian. In the first half of the talk, we will discuss a formula linking this operator to a natural Hodge Laplacian on a hypersurface. This result bears intriguing resemblance to the Gauss--Codazzi equation for the scalar curvature. In the second half, we will explain how our formula provides an integrability condition for the Poisson equation associated with the G$_2$-Laplacian in the presence of cohomogeneity one symmetry. Joint work with Timothy Buttsworth (The University of New South Wales) and Stepan Hudecek (The University of Queensland).

On symplectic mean curvature flows

Jun Sun Wuhan University Peoples Rep of China Co-Author(s):

Abstract: The symplectic property is preserved under the mean curvature flow in a KE surface, a flow often referred to as the symplectic mean curvature flow. It has been established that no Type I singularity can occur in such a flow. In this talk, we will first collect the known results on symplectic mean curvature flows. Then we will talk about our recent results on singularity analysis for such a flow. This talk is based on joint work with Jingyi Chen, Xiaoli Han and Jiayu Li.

Existence and regularity of multi-phase mean curvature flow in the hyperbolic space

Yoshihiro Tonegawa Institute of Science Tokyo Japan Co-Author(s):    Qing Han, Nan Wu

Abstract: We report some existence and regularity results on the multi-phase mean curvature flow in the standard hyperbolic space of general dimensions. Under a mild regularity assumption on the initial data, we prove the global-in-time existence and regularity results of the mean curvature flow. In particular, we show that the smoothness of the asymptotic boundary of the mean curvature flow persists for all time if the initial data is smooth. This reveals an interesting time-dependent regularity property different from the static case where the dimension plays an important role.

Uniqueness of self-similar solutions to curvature flows and uniqueness of solutions to isotropic curvature problems

Xianfeng Wang Nankai University Peoples Rep of China Co-Author(s):

Abstract: Self-similar solutions play an important role in the study of the asymptotic behaviors of curvature flows, and are closely related to some prescribed curvature problems. In this talk, we discuss the uniqueness of self-similar solutions to some fully nonlinear curvature flows, as well as the uniqueness of solutions to some isotropic curvature problems.

A Heintze-Karcher-type inequality for capillary hypersurfaces in hyperbolic space

Tailong Zhou Sichuan University Peoples Rep of China Co-Author(s):    Yingxiang Hu, Yong Wei, Chao Xia

Abstract: In this talk we prove a Heintze-Karcher type inequality and a Minkowski type formula for capillary hypersurfaces in hyperbolic space with boundary supported on general totally umbilical hypersurfaces. As an application, we prove an Alexandrov-type theorem for capillary hypersurfaces with constant kth mean curvature supported on totally umbilical hypersurfaces in hyperbolic space.