Organizers: Da Rong Cheng, Greg Galloway and Pengzi Miao

For the Fall 2023 semester, the seminar meets Friday from 4:00 PM to 5:00 PM in Ungar 402 unless otherwise noted.

** Fall 2023 **

**October 20th: ** Marcelo Disconzi (Vanderbilt University)

*"The relativistic Euler equations with a physical vacuum boundary"*

We consider the relativistic Euler equations with a physical vacuum boundary and an equation of state \(p(\rho) = \rho^\gamma\), \(\gamma > 1\). We establish the following results. (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and continuous dependence on the data; (ii) low regularity solutions: our uniqueness result holds at the level of Lipschitz velocity and density, while our rough solutions, obtained as unique limits of smooth solutions, have regularity near scaling; (iii) stability: our uniqueness in fact follows from a more general result, namely, we show that a certain nonlinear functional that tracks the distance between two solutions (in part by measuring the distance between their respective boundaries) is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions; (v) we establish a sharp continuation criterion, at the level of scaling, showing that solutions can be continued as long as the velocity is in \(L^1_t Lip_x\) and a suitable weighted version of the density is at the same regularity level. This is joint work with Mihaela Ifrim and Daniel Tataru.

**October 13th: ** Abraão Mendes (Universidade Federal de Alagoas)

*"Rigidity of min-max minimal disks in 3-balls with non-negative Ricci curvature"*

In this lecture, we are going to present a rigidity statement for free boundary minimal surfaces produced via min-max methods. More precisely, for any Riemannian metric \(g\) on the 3-ball \(B\) with non-negative Ricci curvature and \(\mbox{II}_{\partial B}\geq g_{|\partial B}\), there exists a free boundary minimal disk \(\Delta\) of least area among all free boundary minimal disks in \((B, g)\). Moreover, the area of any such \(\Delta\) equals to the width of \((B, g)\), \(\Delta\) has index one, and the length of \(\partial\Delta\) is bounded from above by \(2\pi\). Furthermore, the length of \(\partial\Delta\) equals to \(2\pi\) if and only if \((B, g)\) is isometric to the Euclidean unit ball. This is related to a rigidity result obtained by F.C. Marques and A. Neves in the closed case. The proof uses a rigidity statement concerning half-balls with non-negative Ricci curvature which is true in any dimension.

** Spring 2023 **

**April 25th: ** Sven Hirsch (Duke University)

*"On a generalization of Geroch's conjecture"*

The theorem of Bonnet-Myers implies that manifolds with topology \(M^{n-1}\times S^1\) do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture shows that the torus \(T^n\) does not admit a metric of positive scalar curvature. In this talk I will introduce a new notion of curvature which interpolates between Ricci and scalar curvature (so-called \(m\)-intermediate curvature) and use stable weighted slicings to show that for \(n\leq 7\) the manifolds \(M^{n-m}\times T^m\) do not admit a metric of positive \(m\)-intermediate curvature. This is joint work with Simon Brendle and Florian Johne.

**April 18th (virtual talk): ** Abraão Mendes (Universidade Federal de Alagoas)

*"Some rigidity results for compact initial data sets"*

In this lecture, we aim to present some rigidity results for compact initial data sets, in both the boundary and no boundary cases. For example, under natural energy, boundary, and topological conditions, we obtain a global version of a well-known result of H. Bray, S. Brendle, and A. Neves. We also prove an extension of a result obtained, in a previous work, by M. Eichmair, G.J. Galloway, and the author. Finally, as time permits, we are going to present an example in order to illustrate the last result presented in this lecture. This is part of a joint work with G.J. Galloway.

**April 4th: ** Eric Ling (University of Copenhagen)

*"Some new results and open questions related to the cosmological constant appearing in inflationary models"*

The cosmological constant appears as an initial condition for Milne-like spacetimes, a class of hyperboloidal inflationary models. In previous work it was shown that this remains true even if one removes the homogeneous and isotropic assumptions for Milne-like spacetimes. However, examples of such nonhomogeneous versions of Milne-like spacetimes are lacking. In this talk, we suggest some ways of constructing an initial value problem for Milne-like spacetimes which could provide such examples. Lastly, in joint work with Ghazal Geshnizjani and Jerome Quintin, we show that the cosmological constant also appears as an initial condition for a class flat FLRW models which are asymptotically de Sitter towards the past.

**March 7th: ** Conghan Dong (Stony Brook University)

*"Stability of Euclidean 3-space for the positive mass theorem"*

The Positive Mass Theorem of R. Schoen and S.-T. Yau in dimension 3 states that if \((M^3, g)\) is asymptotically flat and has nonnegative scalar curvature, then its ADM mass \(m(g)\) satisfies \(m(g) \geq 0\), and equality holds only when \((M, g)\) is the flat Euclidean 3-space \(\mathbb{R}^3\). We show that \(\mathbb{R}^3\) is stable in the following sense. Let \((M^3_i, g_i)\) be a sequence of asymptotically flat 3-manifolds with nonnegative scalar curvature and suppose that \(m(g_i)\) converges to 0. Then for all \(i\), there is a domain \(Z_i\) in \(M_i\) such that the area of the boundary \(\partial Z_i\) converges to zero and the sequence \((M_i \setminus Z_i , \hat{d}_{g_i} , p_i )\), with induced length metric \(\hat{d}_{g_i}\) and any base point \(p_i \in M_i \setminus Z_i\), converges to \(\mathbb{R}^3\) in the pointed measured Gromov-Hausdorff topology. This confirms a conjecture of G. Huisken and T. Ilmanen. We also find an almost optimal bound for the area of \(\partial Z_i\) in terms of \(m(g_i)\). This is joint work with Antoine Song.

**February 14th: ** Xu Cheng (Instituto de Matematica e Estatistica, Universidade Federal Fluminense)

*"Volume of hypersurfaces in \(\mathbb{R}^n\) with bounded weighted mean curvature"*

In this talk, we will discuss the volume property of complete noncompact submanifolds in a gradient shrinking Ricci soliton with bounded weighted mean curvature vector. Roughly speaking, such a submanifold must have polynomial and at least linear volume growth. An example is properly immersed complete noncompact hypersurfaces in \(\mathbb{R}^n\) with bounded Gaussian-weighted mean curvature, including self-shrinkers. This is a joint work with M. Vieira and D. Zhou.

**January 31st: ** Detang Zhou (Instituto de Matematica e Estatistica, Universidade Federal Fluminense)

*"Rigidity of 4-dimensional Shrinking Ricci solitons"*

Perelman defined his \(W\)-functional and proved the entropy monotonicity formulae for Hamilton's Ricci flow. The critical points of \(W\)-functional are shrinking gradient Ricci solitons (SGRS). It is well known that gradient Ricci solitons are generalizations of Einstein manifolds and basic models for smooth metric measure spaces. In this talk I will discuss some recent progress and problems in four dimensional cases. In particular, one of the challenging problems is to classify all gradient Ricci solitons with constant scalar curvature. Recently in a joint work with X. Cheng, we prove that a 4-dimensional shrinking gradient Ricci soliton has constant scalar curvature if and only if it is either Einstein, or a finite quotient of Gaussian shrinking soliton \(\mathbb{R}^4\), \(\mathbb{S}^2 \times \mathbb{R}^2\) or \(\mathbb{S}^3 \times \mathbb{R}\).

** Fall 2022 **

**December 2nd (virtual talk, 2:00 PM): ** Thomas Körber (University of Vienna)

*"The Riemannian Penrose inequality for asymptotically flat half-spaces and rigidity"*

Asymptotically flat half-spaces \((M,g)\) are asymptotically flat manifolds with a non-compact boundary. They naturally arise as suitable subsets of initial data for the Einstein field equations. In this talk, I will present a proof of the Riemannian Penrose inequality for asymptotically flat half-spaces with horizon boundary (joint with M. Eichmair) that works in all dimensions up to seven. This inequality gives a sharp bound for the area of the horizon boundary in terms of the half-space mass of \((M,g)\). To prove the inequality, we double \((M, g)\) along its non-compact boundary and smooth the doubled manifold appropriately. To prove rigidity, we use variational methods to show that, if equality holds, the non-compact boundary of \((M,g)\) must be totally geodesic. I will also explain how our techniques can be used to prove rigidity for the Riemannian Penrose inequality for asymptotically flat manifolds.

**November 18th (virtual talk): ** Pengzi Miao (University of Miami)

*"Positive harmonic functions in 3-dimension"*

I will discuss some new properties of positive harmonic functions in dimension three. Applications include families of inequalities relating the surface capacity, Willmore functional, and the mass of asymptotically flat 3-manifolds. A by-product shows additional proofs of the 3-dimensional Riemannian positive mass theorem.

The material of this talk will be a subset of the paper https://arxiv.org/abs/2207.03467.

**November 11th: ** Xiaoxiang Chai (Korea Institute for Advanced Study)

*"Band width estimates in CMC initial data sets and applications"*

Gromov showed that a \(n\) dimensional toroical band with lower scalar curvature bound \(n(n-1)\), the distance of two boundary components of the band is bounded below by \(\pi/n\). There are various generalizations of this band width estimate. We provide a generalization to the spacetime settings. In particular, we study the band width estimate torical band which is also a CMC initial data set. We give a proof using a hypersurface of prescribed null expansion and discuss other proofs. We apply this band width estimates to study the positive mass theorem for asymptotically hyperbolic manifolds with arbitrary ends. This is based joint works of Xueyuan Wan (Chongqing University of Technology).

**November 4th (virtual talk): ** Abraão Mendes (Universidade Federal de Alagoas)

*"Classification of exterior free boundary minimal hypersurfaces"*

In this lecture we aim to present two classification theorems for exterior free boundary minimal hypersurfaces (exterior FBMH for short) in Euclidean space. The first result states that the only exterior stable FBMH with parallel embedded regular ends are the catenoidal hypersurfaces. To achieve this we first prove a Bocher-type result for positive Jacobi functions on regular minimal ends in \(\mathbb{R}^{n+1}\) which, after some calculations, implies the first theorem. The second theorem states that any exterior FBMH \(\Sigma\) with one regular end is a catenoidal hypersurface. Its proof is based on a symmetrization procedure due to R. Schoen. Finally, we give a complete description of the catenoidal hypersurfaces, including the calculation of their indices. This lecture is based on a joint work with L. Mazet.

**October 28th (virtual talk): ** Gaoming Wang (Cornell University)

*"Second order elliptic operators on triple junction surfaces"*

In this talk, we will consider minimal triple junction surfaces, a special class of singular minimal surfaces whose boundaries are identified in a particular manner. Hence, it is quite natural to extend the classical theory of minimal surfaces to minimal triple junction surfaces. Indeed, we can show that the classical PDE theory holds on triple junction surfaces. As a consequence, we can prove a type of Generalized Bernstein Theorem and give the definition of Morse index on minimal triple junction surfaces.

**October 21st (virtual talk): ** Kwok-Kun Kwong (University of Wollongong)

*"Effect of the average scalar curvature on Riemannian manifolds"*

The well-known Bishop-Gromov volume comparison theorem says that if the Ricci curvature is bounded below by \((n-1)k\), then the volume of a metric ball is at most that of the volume of the ball with the same radius in the space form with curvature \(k\). Counterexamples show that the Ricci curvature cannot be replaced by the scalar curvature in the assumption. On the other hand, a Taylor series computation shows that the scalar curvature does tend to decrease the volume of small geodesic balls. In this talk, I will illustrate how the average scalar curvature (together with the Ricci curvature) of a closed manifold affects the average volume of its metric balls of any size. This gives an improvement of the Bishop-Gromov estimate. I will also show its effect on the average total mean curvature of geodesic spheres of radius up to the injectivity radius.

**September 23rd: ** Da Rong Cheng (University of Miami)

*"Existence of Free Boundary Constant Mean Curvature (CMC) Disks"*

Given a surface S in R3, a classical problem is to find disk-type surfaces with prescribed constant mean curvature whose boundary meets S orthogonally. When S is diffeomorphic to a sphere, direct minimization could lead to trivial solutions and hence min-max constructions are needed. Among the earliest such constructions is the work of Struwe, who produced the desired free boundary CMC disks for almost every mean curvature value up to that of the smallest round sphere enclosing S. In a joint work with Xin Zhou (Cornell), we combined Struwe's method with other techniques to obtain an analogous result for CMC 2-spheres in Riemannian 3-spheres and were able to remove the "almost every" restriction in the presence of positive ambient curvature. In this talk, I will report on more recent progress where the ideas in that work are applied back to the free boundary problem to refine and improve Struwe's result.