University of Tennessee - Geometric analysis seminar

The seminar will be run via Zoom (ID: 912 2826 6699).

Please contact tbourni at utk dot edu to sign up for the mailing list.

Fall 2021

Tuesday, September 7, 1450-1550
Zhongshon An (University of Connecticut)
Title: Static vacuum extensions of Bartnik boundary data near flat domains.
Abstract: The Bartnik boundary data of a Riemannian manifold with nonempty boundary consists of the induced metric and extrinsic mean curvature of the boundary manifold. Existence of static vacuum Riemannian metrics with prescribed Bartnik data is one of the most fundamental problems in Riemannian geometry related to general relativity. It is also a very interesting problem on the global solvability of a natural geometric boundary value problem. In this talk I will first discuss the basic properties of static vacuum metrics and their boundary geometry. Then I will present some recent progress towards the existence problem based on a joint work with Lan-Hsuan Huang.

Tuesday, September 14, 1450-1550
Sathya Rengaswami (University of Tennessee) [Live]
Title: Translating solutions to extrinsic geometric flows.
Abstract: Analogous to the bowl soliton of mean curvature flow, we construct convex rotationally symmetric translating solutions to a very large class of flow speeds, namely those that are positively homogeneous, elliptic and symmetric with respect to the principal curvatures. We give precise criteria which detect when these translators are defined on all of Rn or contained in a cylinder. For speeds that are nonzero when at least one of the principal curvatures is nonzero, we also describe the asymptotics of the translator at infinity.

Tuesday, September 21, 1450-1550
Melanie Rupflin (University of Oxford)
Title: Łojasiewicz inequalities near simple bubble trees.
Abstract: In the study of (almost) critical points of a geometric variational problem one is often confronted with the problem that a weakly-obtained limiting object does not have the same topology. A major challenge in such situations is that the seminal results of Simon on Łojasiewicz inequalities, one of the most powerful tools in the analysis of the energy spectrum of analytic energies and the corresponding gradient flows, are not applicable. In this talk we present a method that allows us to prove such Łojasiewicz inequalities also when the topology changes e.g. due to the formation of a bubble and present applications for the H-surface energy and the harmonic map heat flow.

Tuesday, September 28, 1450-1550
Ben Lambert (University of Derby)
Title: Lagrangian mean curvature flow with boundary.
Abstract: The foundational result of Lagrangian mean curvature flow (LMCF) is that, in Calabi–Yau manifolds, high codimensional mean curvature flow preserves the Lagrangian condition. A natural question is then to ask if this can this be generalised to manifolds with boundary. Equivalently, what is a well-defined boundary condition for LMCF? In this talk I will provide an answer to this question, and then demonstrate that the resulting flow exhibits good behaviour in two model situations, namely with boundary on the Lawlor neck and Clifford torus, respectively. No prior knowledge of geometric flows will be assumed. This work is joint with Chris Evans and Albert Wood.

Tuesday, October 19, 1450-1550
Alex Mramor (Johns Hopkins University) [Live]
Title: Some new applications of the mean curvature flow to self shrinkers
Abstract: The mean curvature flow, where one deforms a submanifold by its mean curvature vector, is known to in many cases develop singularities. These are points where the curvature along the flow blows up, or in some sense where the submanifold pinches. This makes the study of singularities vital to fully utilize the flow. Arguably the most basic local models for singularities are self shrinkers, called such because they evolve by dilations. In this talk I’ll discuss some applications of the mean curvature flow to self shrinkers in R3 and R4.

Tuesday, October 26, 1450-1550
Simon Blatt (Paris-Lodron University Salzburg)
Title: Analyticity of solutions to fractional partial differential equations.
Abstract: We will discuss an old topic in the field of partial differential equations in a new context: The question of analyticity of solutions to elliptic equations. While first results for classical elliptic partial differential equations were already obtained by Bernstein in 1904, in the context of fractional and non-local equations only partial results or results for very special cases like the Hartree–Fock equations and the Boltzmann equation are known up to now. After presenting some known results, we will discuss our recent findings for so-called knot energies and general semi-linear integro-differential equations. The main ingredients in the proof of these results are Cauchy's method of majorants and a new estimate for the long range interactions of these equations. Partly joint with Nicole Vorderobermeier.

Tuesday, November 2, 1450-1550 [Moved to December 1]
Felix Schulze (University of Warwick)
Title: Mean curvature flow with generic initial data.
Abstract: Mean curvature flow is the gradient flow of the area functional and constitutes a natural geometric heat equation on the space of hypersurfaces in an ambient Riemannian manifold. It is believed, similar to Ricci flow in the intrinsic setting, to have the potential to serve as a tool to approach several fundamental conjectures in geometry. The obstacle for these applications is that the flow develops singularities, which one in general might not be able to classify completely. Nevertheless, a well-known conjecture of Huisken states that a generic mean curvature flow should have only spherical and cylindrical singularities. As a first step in this direction Colding–Minicozzi have shown in fundamental work that spheres and cylinders are the only linearly stable singularity models. As a second step toward Huisken's conjecture we show that mean curvature flow of generic initial closed surfaces in R3 avoids asymptotically conical and non-spherical compact singularities. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact self-similarly shrinking solutions. This is joint work with Otis Chodosh, Kyeongsu Choi and Christos Mantoulidis.

Tuesday, November 9, 1450-1550
Magdalena Rodriguez (Universidad de Granada)
Title: Constant mean curvature surfaces in H2×R.
Abstract: The theory of constant mean curvature H>0 surfaces (H-surfaces) in H2×R became very active after the seminal work by Abresch and Rosenberg where they described a Hopf-type holomorphic quadratic differential on any such surface and classified the rotational H-spheres. The critical value for H in H2×R is 1/2, in the sense that there exist compact examples only when H>1/2 and entire graphs if H≤1/2. When H>1/2, the geometric behaviour of the H-surfaces in H2×R is analogous, in some sense, to the surfaces of positive constant mean curvature in R3. In this talk we will prove that a properly embedded H-surface in H2×R with 0<H≤1/2 cannot be contained in a horizontal slab if it has an annular end. Moreover we will show that, when 0<H≤1/2, a properly embedded H-surface with finite topology contained in H2×[0,∞) must be a graph. This is a joint work with Laurent Hauswirth and Ana Menezes.

Tuesday, November 26, 1450-1550
Julian Scheuer (Cardiff University)
Title: The mean curvature flow in null hypersurfaces and the detection of MOTS.
Abstract: This talk is based on joint work with Henri Roesch. We discuss the mean curvature flow in 3-dimensional null hypersurfaces. In a spacetime a hypersurface is called null, if its induced metric is degenerate. The speed of the mean curvature flow of spacelike surfaces in a null hypersurface is the projection of the codimension-two mean curvature vector onto the null hypersurface. Under fairly mild conditions we obtain that for an outer un-trapped initial surface, a condition which resembles the mean-convexity of a surface in Euclidean space, the mean curvature flow exists for all times and converges smoothly to a marginally outer trapped surface (MOTS). As an application we obtain the existence of a smooth local foliation of the past of an outermost MOTS.

Tuesday, November 23, 1450-1550
Xuan Hien Nguyen (University of Iowa) [Live]
Title: The fundamental gap of horoconvex domains in hyperbolic space.
Abstract: In this talk, we introduce the fundamental gap problem and its characteristics in hyperbolic space. We will spend some time recalling the special properties of hyperbolic geometry before showing that, unlike in the Euclidean or spherical cases, the product of their fundamental gap with the square of their diameter has no positive lower bound. The result follows from the study of the fundamental gap of geodesic balls as the radius goes to infinity, then comparing horoconvex domains to balls.

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