University of Tennessee - Geometric analysis seminar

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Spring 2021

Tuesday, February 9, 14:50
Sathya Rengaswami (UTK)
Title: Rotationally symmetric translators of curvature flows I.
Abstract: We construct and analyse rotationally symmetric translating solutions to a large class of geometric flows.

Tuesday, February 16, 14:50
Sathya Rengaswami (UTK)
Title: Rotationally symmetric translators of curvature flows II.
Abstract: We construct and analyse rotationally symmetric translating solutions to a large class of geometric flows.

Tuesday, February 23, 14:50
Stephen Lynch (Tübingen)
Title: Convex ancient solutions of mean curvature flow with type I curvature growth.
Abstract: Solutions of mean curvature flow that exist for all negative times (ancient solutions) are of fundamental importance, in part because they arise as blow-up limits at singularities. We will show that every convex ancient solution of type I (meaning that its curvature grows like that of a shrinker) is in fact a family of shrinking cylinders. The argument does not make use of Huisken's monotonicity formula, and so can be applied to a wide range of fully nonlinear flows as well.

Tuesday, March 2, 14:50
Mario Santilli (Augsburg)
Title: Soap bubble theorems in convex geometry and geometric measure theory.
Abstract: The celebrated Alexandrov theorem (1958) asserts that a compact and embedded smooth hypersurface with constant mean curvature in the Euclidean space must be a round sphere. In this talk I will discuss two recent extensions of this result to singular varieties. The first extension deals with sets of finite perimeter with constant (possibly anisotropic) distributional mean curvature, which correspond to the critical points of the (possibly anisotropic) isoperimetric functional. The second extension deals with arbitrary convex bodies.

Tuesday, March 09, 14:50
Giuseppe Tinaglia (King's College London)
Title: The geometry of constant mean curvature surfaces in R3.
Abstract: I will begin by reviewing classical geometric properties of constant H>0 mean curvature surfaces in R3. I will then talk about several more recent results for surfaces embedded in R3 with constant mean curvature, such as curvature and injectivity radius estimates (for simply connected surfaces). Finally, I will show applications of such estimates including a characterization of the round sphere as the only simply connected surface embedded in R3 with constant mean curvature, and area estimates for compact surfaces embedded in a flat torus with constant mean curvature and finite genus. This is joint work with Bill Meeks.

Tuesday, March 16, 14:50
Alec Payne (Courant Institute)
Title: Mass drop and multiplicity in mean curvature flow.
Abstract: A generic mean curvature flow can be continued through singularities via two possible weak solutions: Brakke flow and level set flow. Brakke flows may have discontinuous mass over time, i.e. have mass drop, which makes them tantamount to subsolutions to mean curvature flow. On the other hand, level set flows may attain positive measure, making them like supersolutions to the flow. In this talk, we will discuss the discrepancy between these two flows and relate this to various fundamental open problems in mean curvature flow. In particular, we discuss how flows with only generic singularities have no mass drop and satisfy equality in the Brakke inequality. This generalizes Metzger–Schulze's result for mean convex flows. We conclude by discussing the implications of these results for the classification of limit flows.

Tuesday, March 23 14:50
Christos Sourdis (National Kapodistrian University of Athens)
Title: Liouville type theorems for ancient solutions to semi-linear heat equations with critical or supercritical exponent.
Abstract: Firstly, we will provide a survey of Liouville type results for global, ancient or eternal solutions to the nonlinear heat equation ut=Δu+|u|p–1u, p>1, in the whole space. We will then turn our attention to critical or supercritical exponents p with respect to the Sobolev embedding. In that regime, the steady state problem has a continuum of sign-definite radially symmetric solutions, including singular ones, that decay to zero at infinity. We will particularly focus on the case where p is larger or equal to the so called Joseph–Lundgren exponent, in which case the aforementioned radial steady states form a foliation. Our Liouville type results imply the quasiconvergence as t goes to infinity of a class of solutions to the corresponding initial value problem.

Tuesday, March 30, 14:50
Debora Impera (Politecnico di Torino)
Title: Quantitative index bounds for f-minimal hypersurfaces in the Euclidean space.
Abstract: The recent developments in the existence theory for minimal immersions have motivated a renewed interest in studying estimates on the Morse index of these objects. One possible way to control instability is through topological invariants (in particular through the first Betti number) of the minimal hypersurface. This was first investigated by A. Ros for immersed minimal surfaces in R3, or a quotient of it by a group of translations, and then, in higher dimension, by A. Savo when then ambient manifold is a round sphere. In this talk we will first discuss how the method used by Savo can be generalized to study the Morse index of self-shrinkers for the mean curvature flow and, more generally, of weighted minimal hypersurfaces in a Euclidean space endowed with a convex weight. In particular, when the hypersurface is compact, we will show that the index is bounded from below by an affine function of its first Betti number. In the complete non-compact case, the lower bound is in terms of the dimension of the space of weighted square integrable f-harmonic 1-forms. In particular, in dimension 2, the procedure gives an index estimate in terms of the genus of the surface.

Combining this technique with an adaptation to the weighted setting of well-known results by P. Li and L. F. Tam, we will also discuss how to obtain quantitative estimates on the Morse index of translators for the mean curvature flow with bounded norm of the second fundamental form via the number of ends of the hypersurface.

This talk is based on joint works with Michele Rimoldi and Alessandro Savo.

Tuesday, April 6, 14:50
Andreas Savas-Halilaj (University of Ioannina)
Title: Minimal maps, MCF and isotopy problems.
Abstract: I will discuss the mean curvature flow (MCF) of graphical submanifolds generated by smooth maps between Riemannian manifolds. I will demonstrate applications related to the homotopy type of smooth maps between compact manifolds. I will also show some rigidity results concerning the Hopf fibrations.

Tuesday, April 13, 14:50
Panagiotis Gianiotis (National Kapodistrian University of Athens)
Title: Isometric flow of G2 structures.
Abstract: A G2 structure on a 7 manifold is a three form that determines, in a non linear way, a Riemannian metric. Our interest in such structures comes from the fact that when they are parallel with respect to the associated Levi-Civita connection then the metric is automatically Ricci flat with holonomy contained in the Lie group G2. Parallel G2 structures can be considered as the optimal such structures on a given smooth manifold, however they may not exist since there are several obstructions. Unfortunately, despite the construction of many examples of parallel G2 structures, there is at the moment no conjecture regarding which smooth 7 manifolds admit holonomy G2 metrics. On the other hand, any Riemannian metric on a manifold admitting G2 structures is induced by many - isometric - G2 structures, and a natural question is to find whether there exists an optimal representative in a given isometric class. In this talk I will discuss a geometric flow approach to this problem, initially proposed by Grigorian, and present joint work with Dwivedi and Karigiannis in which we develop the foundational theory for this flow.

Tuesday, April 20, 14:50
Jose Espinar (Cadiz University)
Title: On non-compact free boundary minimal hypersurfaces in the Riemannian Schwarzschild spaces.
Abstract: We will show that, in contrast with the 3-dimensional case, the Morse index of a free boundary rotationally symmetric totally geodesic hypersurface of the n-dimensional Riemannnian Schwarzschild space with respect to variations that are tangential along the horizon is zero, for $n\geq 4$. Moreover, we will show that there exist non-compact free boundary minimal hypersurfaces which are not totally geodesic, n≥ 8, with Morse index equal to zero. Also, for n≥4, there exist infinitely many non-compact free boundary minimal hypersurfaces, which are not congruent to each other, with infinite Morse index. Finally, we will discuss the density at infinity of a free boundary minimal hypersurface with respect to a minimal cone constructed over a minimal hypersurface of the unit Euclidean sphere. We obtain a lower bound for the density in terms of the area of the boundary of the hypersurface and the area of the minimal hypersurface in the unit sphere. This lower bound is optimal in the sense that only minimal cones achieve it.

Tuesday, April 27, 14:50
Christian Scharrer (University of Warwick)
Title: The isoperimetric constrained Willmore tori.
Abstract: In order to explain the bi-concave shape of red blood cells, Helfrich proposed the minimisation of a bending energy amongst closed surfaces with given fixed area and volume. In the homogeneous case, the Helfrich functional reduces to the scaling invariant Willmore functional. Thus, for the minimisation, the constraints on area and volume reduce to a single constraint on the scaling invariant isoperimetric ratio. This talk is about two strict inequalities that lead to existence of isoperimetric constrained Willmore tori.

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