# Geometry & Analysis Seminar Spring 2015

Thursday 4:00 - 5:00pm

ROOM 4130 MCHENRY LIBRARY

For further information please contact Professor Longzhi Lin or call 831-459-2969

**Thursday April 2, 2015**

***CANCELLED***

**From a Question of S.-S. Chern to B.-Y. Chen’s Invariants: New Inequalities for Curvature****Bogdan Suceavă, California State University, Fullerton**

In 1968, S.-S. Chern raised to attention the question that there might be further Riemannian obstructions for a manifold to admit an isometric minimal immersion into an Euclidean space. This question triggered a whole new direction of study in the geometry of submanifolds. In the last two decades, there have been important advances in the study of new curvature invariants, many of these efforts due to B.-Y. Chen. We will present a few recent developments in this theory, and we focus our attention on several classes of inequalities in the geometry of submanifolds in Riemannian and Kaehlerian context.

**Thursday April 9, 2015**

**Convexity estimates for mean curvature flow with free bounda****ry**

**Nick Edelen, Stanford University**

We prove the convexity estimates of Huisken-Sinestrari for finite-time singularities of mean-convex, mean curvature flow with free boundary in a barrier $S$, and thereby show any limit flow is weakly convex. Here $S$ can be any properly embedded, oriented surface in $R^{n+1}$ of bounded geometry.

**Closed geodesics are straightest paths in a space that happen to be periodic. (Think great circles in a sphere.) Classical study of the existence of closed geodesics relies on assumptions of both smoothness and topology. For example, a Riemannian (smooth) manifold has a closed geodesic if it is compact without boundary (topologically nice). Counterexamples show that neither assumption can be removed completely. We'll look at the special case of relaxing smoothness to allow certain cone singularities and find corresponding existence results. Surprisingly, we'll see that existence of closed geodesics holds here for some topologies where smoothness everywhere is not enough. (Joint with Longzhi Lin)**

Jonathan Dahl, Lafayette College

**Thursday April 16, 2015****Closed geodesics on smooth surfaces with sharp cone singularities**Jonathan Dahl, Lafayette College

**The theorem in the title is about disk-type solutions of the classical Plateau problem in R^3. The proof will take three lectures.**

Michael Beeson, San Jose State University

**Thursday April 23, 2015**

*A real-analytic Jordan curve cannot bound infinitely many relative minima of area. (Part 1)*Michael Beeson, San Jose State University

By work of Tomi it suffices to refute the existence of a one-parameter family u(t) of solutions of Plateau's problem which are relative minima for t > 0 but u(0) has a branch point.

The hard case is when u(0) has a boundary branch point. We will analyze this situation using the Gauss-Bonnet theorem and the first eigenfunction of the eigenvalue problem associated with the second variation of area, which arises as the normal component of du/dt. What happens for small t in the vicinity of the origin (where the branch point develops when t=0) is the focus of the analysis. There, for small positive t, the Gauss map develops "bubbles" that disappear in the limit; the eigenfunction goes to zero, but its leading term, after blow-up near origin, is a harmonic function satisfying a nice differential equation, that retains the property of the eigenfunction of having just one sign.

Using all this information, the situation is ultimately shown to be impossible.

**Thursday April 30, 2015****The theorem in the title is about disk-type solutions of the classical Plateau problem in R^3. The proof will take three lectures.**

Michael Beeson, San Jose State University

*A real-analytic Jordan curve cannot bound infinitely many relative minima of area. (Part 2)*Michael Beeson, San Jose State University

By work of Tomi it suffices to refute the existence of a one-parameter family u(t) of solutions of Plateau's problem which are relative minima for t > 0 but u(0) has a branch point.

The hard case is when u(0) has a boundary branch point. We will analyze this situation using the Gauss-Bonnet theorem and the first eigenfunction of the eigenvalue problem associated with the second variation of area, which arises as the normal component of du/dt. What happens for small t in the vicinity of the origin (where the branch point develops when t=0) is the focus of the analysis. There, for small positive t, the Gauss map develops "bubbles" that disappear in the limit; the eigenfunction goes to zero, but its leading term, after blow-up near origin, is a harmonic function satisfying a nice differential equation, that retains the property of the eigenfunction of having just one sign.

Using all this information, the situation is ultimately shown to be impossible.

**Thursday May 7, 2015****The theorem in the title is about disk-type solutions of the classical Plateau problem in R^3. The proof will take three lectures.**

Michael Beeson, San Jose State University

*A real-analytic Jordan curve cannot bound infinitely many relative minima of area. (Part 3)*Michael Beeson, San Jose State University

By work of Tomi it suffices to refute the existence of a one-parameter family u(t) of solutions of Plateau's problem which are relative minima for t > 0 but u(0) has a branch point.

The hard case is when u(0) has a boundary branch point. We will analyze this situation using the Gauss-Bonnet theorem and the first eigenfunction of the eigenvalue problem associated with the second variation of area, which arises as the normal component of du/dt. What happens for small t in the vicinity of the origin (where the branch point develops when t=0) is the focus of the analysis. There, for small positive t, the Gauss map develops "bubbles" that disappear in the limit; the eigenfunction goes to zero, but its leading term, after blow-up near origin, is a harmonic function satisfying a nice differential equation, that retains the property of the eigenfunction of having just one sign.

Using all this information, the situation is ultimately shown to be impossible.

**Thursday May 14, 2015**

**The Geometry of Vacuum Static Spaces and Related Topics.****Wei Yuan, University of California, Santa Cruz**

In this talk, I will discuss main results of my Ph.D thesis, which is about the geometric structure of vacuum static spaces and applications of ideas in the study of such spaces to other related geometric problems. It includes: 1. Classifications of vacuum static spaces; 2. Local rigidity phenomena of scalar curvature; 3. Brown-York mass and compactly conformal deformations; 4. Deformations of Q-curvature.

**This will also be Wei Yuan's Ph.D. thesis defense (under the supervision of Professor Jie Qing).**

**Thursday May 21, 2015**

**Periodic brake orbits in the N-body problem****Nai-chia Chen, University of California, Santa Cruz**

This talk is devoted to finding periodic brake orbits in the N-body problem. We consider certain subsystems of the N-body problem that have two degrees of freedom, including the isosceles three-body problem and other highly symmetric sub-problems. We prove the existence of several families of periodic brake orbits by using topological shooting arguments.

***please note date change*****Friday May 22, 2015***Some applications of time derivative bound to Ricci flow***Qi Zhang, University of California, Riverside**

We present a joint work with Richard Bamler. We consider Ricci flows that satisfy certain scalar curvature bounds. It is found that the time derivative for the solution of the heat equation and the curvature tensor have better than expected bounds. Based on these, we derive a number results. They are: bounds on distance distortion at different times and Gaussian bounds for the heat kernel, backward pseudolocality, $L2$-curvature bounds in dimension $4$.

**The Jacobi-Maupertuis principle formulates mechanics at a fixed energy level as finding geodesics of a certain metric. The sign of curvature associated to this metric can then have dynamical consequences. This talk is about results on the signs of these sectional curvatures in the 3 and 4 body case with a strong force and some consequences.**

Connor Jackman, University of California, Santa Cruz

**Thursday May 28, 2015**

*Fitting Pants to N-body problems*Connor Jackman, University of California, Santa Cruz