Speakers and Abstracts

Title: A scalar-mean curvature extremality result for spaces with asymptotically conical singularities

Time: 8am(Rome)==2am(US East)==2pm(China) 

Abstract: We use the Dirac operator method to prove a scalar-mean curvature comparison theorem for spin manifolds which carry iterated conical singularities. Our approach is to study the index theory of a twisted Dirac operator on such singular manifolds. A dichotomy argument is used to prove the comparison theorem without knowing precisely the index of the twisted Dirac operator. This framework also enables us to prove a rigidity theorem of Euclidean domains and a spin positive mass theorem for asymptotically flat manifolds with iterated conical singularities. 

Title: Singular metrics with nonnegative scalar curvature and RCD

Time: 9am(Rome)==3am(US East)==3pm(China) 

Abstract: The existence problem of smooth Riemannian metrics with positive scalar curvature has been well studied and many important results have been established. The studies of weak notions of positive scalar curvature motivate the studies of these problems for singular metrics. In a recent joint work with Prof. Xianzhe Dai, Prof. Lihe Wang and Prof. Guofang Wei, we obtain a Geroch type result for uniformly Euclidean metrics, including a smooth extension result for uniformly Euclidean metric with nonnegative scalar curvature. In particular, this confirms Schoen’s conjecture for isolated singularity on connected sums of torus and compact manifolds. The theory of RCD spaces plays an important role in this work, and we show that a uniformly Euclidean metric smooth away from a singular set of codimension 4 or higher and with Ricci curvature bounded below by K on the smooth part is RCD(K, n).

Title: Foliation of area minimizing hypersurfaces in asymptotically flat manifolds and Schoen's conjecture.

Time: 10am(Rome)==4am(US East)==4pm(China)

Abstract: We present a foliation structure for asymptotically flat manifolds of dimension 4 ≤ n ≤ 7, each leaf being an area minimizing hypersurface. As an application, we prove a drift-to-infinity property of free boundary hypersurfaces in large cylinders lying in asymptotically flat manifolds with nonnegative scalar curvature and positive mass. This is joint work with Yuguang Shi and Haobin Yu.

Title: A result on scalar curvature rigidity 

Time: 11am(Rome)==5am(US East)==5pm(China)

Abstract: We prove that there exist geodesic balls on unit n-sphere, n greater or equal to 3, with sufficiently small radii that are rigid under smooth deformations (possibly large) that increase the scalar curvature and preserve the intrinsic geometry and mean curvature of the boundary. 

Title: Llarull type rigidity for compact domains in a warped product

Time: 1pm(Rome)==7am(US East)==7pm(China) 

Abstract: In 1998, Llarull established a rigidity result for metrics on the round sphere under scalar curvature and metric comparison. In this talk, we report on recent progress toward understanding the Llarull type scalar curvature rigidity for compact domains in a warped product, where the warping function satisfies a log-concavity condition. We formulate a condition on the boundary which is analogous to the log-concavity condition of the warping function. Our main tools are stable prescribed mean curvature surfaces with varying boundary contact angles (also known as stable capillary mu-bubbles) and spinors. This talk is based on joint works with Xueyuan Wan (CQUT) and Gaoming Wang (YMSC).

Title: Stability Theorems for Spheres

Time: 2pm(Rome)==8am(US East)== 8pm(China)

Abstract: In this talk, I'll discuss some recent and ongoing work about the stability of min-max widths of spheres under various lower curvature bounds. Some of this is joint with Davi Maximo, and some with Paul Sweeney Jr. 

Title: The Spacetime Penrose Inequality With Suboptimal Constant

Time: 3pm(Rome)=9am(US East)==9pm(China) 

Abstract: In this talk we will discuss the spacetime Penrose inequality formulated in the context of initial data sets. According to the arguments of Penrose, any violation of this inequality would imply a counterexample to the weak cosmic censorship conjecture, thereby undermining the foundations of general relativity as a deterministic physical theory. By taking advantage of the Jang equation, spacetime harmonic functions, and an inequality established by Conghan Dong and Antoine Song, we are able to establish the spacetime Penrose inequality with suboptimal constant. This is joint work with Edward Bryden, Demetre Kazaras, and Marcus Khuri.

Title: ADM mass, Area and Capacity.

Time: 8am(Rome)==2am(US East)==2pm(China)

Abstract: We prove a sharp mass-capacity inequality and a sharp area-capacity inequality. The first was originally derived by Bray, and the second by Bray and Miao. Our argument relies on two monotone quantities that hold along the level sets of a harmonic function defined in terms of the 2-capacitary potential of the boundary in three-dimensional, complete, one-ended asymptotically flat manifolds with compact, connected boundary and nonnegative scalar curvature, under appropriate assumptions on the topology and the mean curvature of the boundary. If we have time, other sharp comparison inequalities will be introduced.

Title: The Penrose inequality in extrinsic geometry

Time: 9am(Rome)==3am(US East)==3pm(China) 

Abstract: The Riemannian Penrose inequality is a fundamental result in mathematical relativity. It has been a long-standing conjecture of G. Huisken that an analogous result should hold in the context of extrinsic geometry. In this talk, I will present recent joint work with M. Eichmair that resolves this conjecture: The exterior mass of an asymptotically flat support surface S with nonnegative mean curvature is bounded in terms of the area of the outermost free boundary minimal surface supported on S. If equality holds, then the exterior surface of S is a half-catenoid. In particular, we obtain a new characterization of the catenoid among all complete embedded minimal surfaces with finite total curvature. To prove this result, we study minimal capillary surfaces supported on S that minimize the free energy and discover a quantity associated with these surfaces that is nondecreasing as the contact angle increases. 

Title: Quantitative Rigidity for Ricci Curvature

Time: 10am(Rome)==4am(US East)==4pm(China)

Abstract: On manifolds with nonnegative Ricci curvature, we establish a quantitative relationship between the pinching of a monotone functional defined by Colding and the distance to the nearest cone. Moreover, we demonstrate that the pinching quantitatively controls almost splitting. This is joint work with Christine Breiner.

Title: Scalar curvature comparison and rigidity

Time: 11am(Rome)==5am(US East)==5pm(China)

Abstract: For a compact Riemannian 3-manifold (M3, g) with mean convex boundary which is diffeomorphic to a weakly convex compact domain in  ℝ3, we prove that if scalar curvature is nonnegative and the scaled mean curvature comparison H2g ≥ (H0)2gEucl holds then (M,g) is flat. Our result is a smooth analog of Gromov's dihedral rigidity conjecture and an effective version of extremity results on weakly convex balls in  ℝ3. More generally, we prove the comparison and rigidity theorem for several classes of manifolds with corners. Our proof uses capillary minimal surfaces with prescribed contact angle together with the construction of foliation with nonnegative mean curvature and with prescribed contact angles. 

Title: Rigidity for scalar curvature

Time: 1pm(Rome)==7am(US East)==7pm(China)

Abstract: We discuss several rigidity questions for scalar curvature. This includes the resolution of two questions concerning PSC fill-ins by Gromov and Miao, and a geometric characterization of pp-waves. 

Title: Fill-in problems with scalar curvature constraints.

Time: 2pm(Rome)==8am(US East)==8pm(China)

Abstract: A central problem in differential geometry is understanding how the geometry of a boundary determines the geometry of its interior. Gromov's fill-in problem suggests that when a closed Riemannian manifold is filled with a region of large curvature, the extrinsic curvature of the boundary must be bounded above in some sense. The fill-in problem, particularly in the context of scalar curvature, is closely related to certain notions of quasi-local mass in general relativity. In this talk, I will discuss recent progress on the scalar curvature fill-in problem for the torus, which builds upon Brendle-Hung's resolution of the Horowitz-Myers conjecture.

Title: Bounding topological invariants using scalar curvature and isoperimetric constant

Time: 3pm(Rome)==9am(US East)==9pm(China)

Abstract: Scalar curvature encodes the volume information of small geodesic balls within a Riemannian manifold, making it, to some extent, the weakest curvature invariant. This raises a natural question: what topological constraints does scalar curvature impose on manifolds? In this talk, we will show that for a manifold with a scalar curvature lower bound, certain topological invariants can be controlled by its volume and isoperimetric constant. This is joint work with Jinmin Wang, Guoliang Yu, Bo Zhu.

Title: Drawstrings and the geometry of scalar curvature

Time: 8am(Rome)==2am(US East)==2pm(China) 

Abstract: In this talk, we will discuss a recently discovered phenomenon called drawstrings, along with its effects on the geometry of scalar curvature. Roughly speaking, this phenomenon allows one to collapse any codimension 2 submanifold while only decreasing the scalar curvature by ε. This talk is based on joint works with Demetre Kazaras and also Antoine Song

Title: Higher index theory and its applications in scalar curvature

Time: 9am(Rome)==3am(US East)==3pm(China) 

Abstract: The higher index theory is a vast generalization of the classical Atiyah-Singer index theory to non-compact manifolds. The higher index of differential operators is defined in terms of the K-theory of operator algebras, within the framework of Alain Connes' noncommutative geometry. In this talk, I will sketch the construction of the higher index and its application of estimating the bottom spectrum of Laplacian in terms under scalar curvature lower bound. This talk is based on joint work with Bo Zhu. 

Title: 3-Manifolds with positive scalar curvature and bounded geometry 

Time: 10am(Rome)==4am(US East)==4pm(China)

Abstract: We show that a complete contractible 3-manifold with positive scalar curvature and bounded geometry must be ℝ3. We also show that an open handlebody of genus larger than 1 does not admit complete metrics with positive scalar curvature and bounded geometry. This is joint work with O. Chodosh and K. Xu. (See https://arxiv.org/abs/2502.09727)

Title: Decomposition of complete 3-manifolds of positive scalar curvature with quadratic decay

Time: 11am(Rome)==5am(US East)==5pm(China) 

Abstract: A fundamental question in the study of 3-dimensional manifolds consists in understanding the topological structure of 3-manifolds that admit a Riemannian metric of positive scalar curvature, known as PSC manifolds. In the late 1970s, results by Schoen and Yau based on the theory of minimal surfaces and, in parallel, techniques based on the Dirac operator method developed by Gromov and Lawson, led to the classification of closed orientable PSC 3-manifolds: they are precisely those that decompose as a connected sum of spherical manifolds and 𝕊2×𝕊1 summands.

I will present a decomposition result for non-compact PSC 3-manifolds. If a complete Riemannian 3-manifold has positive scalar curvature with subquadratic decay at infinity, then it decomposes as a possibly infinite connected sum of spherical manifolds and 𝕊2×𝕊1. We will also discuss the optimality of this result, which generalises a recent theorem of Gromov and Wang using a more topological approach.

It is a joint work with Florent Balacheff and Stéphane Sabourau.

Title: Asymptotic volume growth ratio on 3-manifolds with positive scalar curvature

Time: 1pm(Rome)==7am(US East)==7pm(China)

Abstract: It was conjectured by Gromov in 1980s that the (n-2)-order asymptotic volume growth ratio for geodesic balls on n-manifolds with positive scalar curvature and nonnegative Ricci curvature should be finite. In 2022, Bo Zhu proved that the ratio is finite with some extra hypotheses. And for dimension 3, there are proofs by Ovidiu Munteanu and Jiaping Wang in 2022, and later by Otis Chodosh, Chao Li and Douglas Stryker in 2023. Using Cheeger-Colding’s almost splitting theorem and the µ-bubble method, we obtained the sharp upper bound for this ratio in dimension 3, and the corresponding rigidity. This is a joint work with Guodong Wei and Guoyi Xu. Some related topics are also included in the talk.

Title: Recent progress on fourth order Willmore energy

Time: 2pm(Rome)==8am(US East)==8pm(China)

Abstract: We introduce a fourth-order Willmore-type problem for closed four-dimensional submanifolds immersed in $R^n$ and establish a connected sum energy reduction for the general fourth-order Willmore energy, analogous to the seminal result of Bauer and Kuwert. We will also discuss recent progress on the study of energy quantization on this energy.

Title: On the topology of stable minimal surfaces in PSC 4-manifolds

Time: 3pm(Rome)==9am(US East)==9pm(China)

Abstract: It is known that a closed 2-sided stable minimal hypersurface in a 4-manifold with positive scalar curvature (PSC) must be Yamabe positive, and hence it is diffeomorphic to a connected sum of S^1 x S^2’s and spherical space forms.  We show that using a new compactness result for minimal surfaces in covering spaces and techniques from 4-manifold topology, one can obtain further control of the topology of stable minimal hypersurfaces. As an application, we show that the outermost apparent horizons of a smooth, asymptotically flat manifold with nonnegative scalar curvature must be diffeomorphic to S^3 or connected sums of S^1 x S^2’s.  This is an extension of Hawking’s black hole topology theorem to dimension 4. The talk is based on joint work with Chao Li.