## Mean Curvature Flow – May 29 – June 1, 2018

### Speakers

- Brian Allen
- Sigurd Angenent
- Jacob Bernstein
- Christine Guenther
- Robert Haslhofer
- Bruce Kleiner (Bradley Lecture)
- Brett Kotschwar
- William Minicozzi
- Xuan Hien Nguyen
- Natasa Sesum
- Organizing Committee
- Theodora Bourni
- Mathew Langford

**INTRODUCTORY WORKSHOP:**

Speaker: Brian Allen

Title: Introducing Mean Curvature Flow

Abstract: We will start by defining Mean Curvature Flow (MCF) and looking at many important examples. Then we will move on to discuss the avoidance principle, scaling properties, evolution equations and first consequences, convergence to a round point for convex solutions, Huisken’s monotonicity formula, singularity analysis , etc. Emphasis will be on giving proof ideas rather than showing all of the details in order to be able to survey as many important properties of MCF as we can at a comfortable pace.

Speaker: Jacob Bernstein

Title: Self-similar solutions to mean curvature flow

Abstract: Self-singular solutions of a geometric heat flow model the singularity formation and resolution of the flow. I will survey some known results as well as some open problems about self-similar solutions of the mean curvature flow- with a particular emphasis on properties of self-shrinking solutions.

SURVEY LECTURES:

Speaker: Sigurd Angenent (2 talks)

Title: 1. Ancient solutions to MCF; 2. MCF from cones

Abstract: The first talk will be on recent joint work with Sesum and Daskalopoulos on ancient convex mean curvature flows; the second talk will discuss non uniqueness results for mean curvature flows from cones.

Speaker: William Minicozzi (2 talks)

Title: Geometric Flows

Abstract: I will talk about recent results on geometric flows.

Speaker: Natasa Sesum (2 talks)

Title: Ancient solutions to geometric flows

Abstract: We will discuss ancient solutions in different geometric flows, the emphasis will be on the mean curvature flow. We will talk about classification results and geometric properties of ancient solutions.

**BRADLEY LECTURE:**

Speaker: Bruce Kleiner

Title: Ricci flow through singularities, diffeomorphism groups, and the Generalized Smale Conjecture.

Abstract: In his resolution of the Poincaré and Geometrization Conjectures, Perelman proved that there is a Ricci flow with surgery starting from any given compact Riemannian 3-manifold. Recognizing that his construction is non-canonical because the surgery procedure depends on a number of arbitrary choices, Perelman conjectured the existence of a canonical Ricci flow through singularities, which is obtainable as a limit of a family of Ricci flows with surgery whose surgery scales tend to zero. The lecture will discuss the solution of Perelman’s conjecture, and an application of Ricci flow through singularities to the Generalized Smale Conjecture, a conjecture about the topology of diffeomorphism groups of 3-manifolds dating from the 1970s.

This is joint work with Richard Bamler

**INVITED TALKS:**

Speaker: Christine Guenther

Title: The Second Order Renormalization Group Flow

Abstract: The Ricci flow appears in physics as the *first order* approximation of the renormalization group flow that arises from the nonlinear sigma model in quantum field theory. In this presentation we introduce the fully nonlinear* second order* flow and compare it with the Ricci flow, considering for example scaling properties, parabolicity, and special solutions. We further present a new entropy for the flow.

Speaker: Robert Haslhofer

Title: Minimal two-spheres in three-spheres

Abstract: We prove that any manifold diffeomorphic to S^3 and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in 1983. Our approach combines ideas from min-max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in three-manifolds. Finally, we apply our methods to solve a problem posed by S.T. Yau in 1987, and to show that the assumptions in the multiplicity one conjecture and the equidistribution of widths conjecture are in a certain sense sharp. This is joint work with Dan Ketover.

Speaker: Brett Kotschwar

Title: Identifying shrinking solitons by their asymptotic geometries

Abstract: Shrinking solitons are self-similar solutions to the Ricci flow and potential models for the structure of a solution in the neighborhood of a developing singularity. All present evidence suggests that the geometric behavior of a complete noncompact shrinking soliton near infinity is highly constrained.

I will describe some uniqueness results obtained recently with Lu Wang which offer some insight into the extent to which a soliton is determined by its asymptotic geometry.

Speaker: Xuan Hien Nguyen

title: Mean curvature flow and self-similar surfaces

Abstract: For curves in the plane, the curvature at a point measures how fast the direction is changing. If each point moves perpendicularly to the loop at a speed equal to the curvature, the resulting flow is called the curve shortening flow. To visualize this flow, one can think of the loop as the edge of a thin layer of ice floating on water. Corners will round out instantly; skinny offshoots disappear fast; inlets fill in; and flat edges move slowly. If the ice layer is circular, it will shrink while remaining circular and disappear eventually. This last example is called a self-similar solution because its shape does not change under the flow. In higher dimensions, because so many paths go through a point, one considers the mean curvature, which is an average of all the curvatures of all the curves through a point, and defines the mean curvature flow. In this talk, I will explore the properties of the mean curvature flow and some classical results. I will then present more recent development about self-similar surfaces. In particular, I will focus on gluing techniques and talk about the main steps and difficulties for gluing pieces of surfaces in order to construct new self-similar solutions.

**SHORT TALKS:**

Speaker: Ian Adelstein

Title: An extension to the Morse energy gradient flow

Abstract: The uniform energy is a discrete energy function and a finite dimensional approximation to the Morse Energy function. The negative gradient flow of the uniform energy deforms a closed curve within its homotopy class in the direction of maximal decrease of the Morse energy. Closed geodesics are the critical points of this flow, and we develop a technique that allows us to restart this flow at these critical points. We show by example that the restarted flow works to improve the minimizing properties of the associated closed geodesics. This technique can be used to bound the length of the shortest closed geodesic and has applications to systolic geometry.

Speaker: Beomjun Choi

Title: Regularity of Non-compact Inverse Mean Curvature Flow

Abstract: We introduce a new method to derive a lower bound of mean curvature which is independent of initial mean curvature. This maximum principle based argument can be localized for convex cases and can be used to prove existence theorem for complete non-compact convex surfaces, which generalizes recent result of P.Daskalopoulos and G.Huisken.

In this talk, we first show how G.Huisken and T.Ilmanen’s result on regularity of compact star-shaped solutions could be reproduced and then talk about local estimates for the existence of general convex non-compact surfaces without mean curvature assumption. This is a joint work with P.Daskalopoulos.

Speaker: Friederike Dittberner

Title: Area-preserving curve shortening flow

Abstract: This talk is about the enclosed area-preserving curve shortening flow for non-convex embedded curves in the plane. We will show that initial curves with a lower bound of −*π* on the local total curvature stay embedded under the flow and develop no singularities in finite time. Moreover, the curves become convex in finite time and converge exponentially and smoothly to a round circle.

Speaker: Siao-Hao Guo

Title: Analysis of Velázquez’s solution to the mean curvature flow with a type II singularity

Abstract: Velázquez discovered a solution to the mean curvature flow which develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons’ cone at infinity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form. This is a joint work with N. Sesum.

Speaker: Sajjad Lakzian

Title: On a flow tangent to the Ricci flow

Abstract: We consider a geometric flow introduced by Gigli and Mantegazza which, in the case of a smooth compact manifold with a smooth metric, is tangential to the Ricci flow almost-everywhere along geodesics. To study spaces with geometric singularities, we consider this flow in the context of a smooth manifold with a rough metric possessing a sufficiently regular heat kernel. On an appropriate non-singular open region, we provide a family of metric tensors evolving in time and provide a regularity theory for this flow in terms of the regularity of the heat kernel. When the rough metric induces a metric measure space satisfying a Riemannian Curvature Dimension condition, we demonstrate that the distance induced by the flow is identical to the evolving distance metric defined by Gigli and Mantegazza on appropriate admissible points. Consequently, we demonstrate that a smooth compact manifold with a finite number of geometric conical singularities remains a smooth manifold with a smooth metric away from the cone points for all future times. Moreover, we show that the distance induced by the evolving metric tensor agrees with the flow of RCD(K,N) spaces defined by Gigli-Mantegazza. This is a joint work with L. Bandara and M. Munn.

Speaker: Stephen Lynch

Title: Pinched ancient solutions to the high codimension mean curvature flow

Abstract: Huisken and Sinestrari have demonstrated that the only compact, uniformly convex ancient solution to the mean curvature flow of hypersurfaces is the homothetically shrinking sphere. We will prove an analogue of their theorem for solutions of arbitrary codimension which, in place of uniform convexity, makes use of a quadratic pinching condition on the second fundamental form. Andrews and Baker have shown that solutions satisfying this pinching condition at the initial time vanish in round points, and we will see how our results lead to a new proof of their theorem. A key ingredient in all of this will be the observation that pinching of the second fundamental form forces positivity of the full curvature operator. This is joint work with Huy Nguyen.

Speaker: Alexander Mramor

Title: Topological rigidity of compact self shrinkers in R^3

Abstract: (joint w/Shengwen Wang) Inspired by previous work of Lawson, Meeks, Frohman and others in this talk we show that compact self shrinkers in R3 are ambiently isotopic to the standard embeddings of genus g surfaces in R^3 by showing they are Heegard splitings (in the appropriate sense) of R3 and appealing to a deep result of Waldhausen.

Speaker: Shengwen Wang

Title: The level set flow of a hypersurface in R^4 of low entropy does not disconnect.

Abstract: We show that if the entropy of a hypersurface in R^4 has entropy below that of the round cylinder S^2\times R, then the level-set flow starting from it will stay connected. As an application we have a sharp version of forward clearing out lemma for non-fattening level-set flow.

Speaker: Ka Wai Wong

Title: Discrete conformal map from compact surface with genus zero to sphere

Abstract: Discrete conformal maps from genus zero surfaces to a sphere are often obtained through minimization of certain energy functionals (e.g. Dirichlet or Willmore energy) and different geometric flows. There are flows using extrinsic geometry such as the mean curvature flow and those using intrinsic geometry (e.g. discrete Ricci flow or discrete Willmore flow). I will give a brief survey on these different methods.