arXiv research

In this paper, we solve the remaining cases of the boundary Yamabe problem introduced by Escobar in 1992. Indeed, using the bubbles of Brendle-Chen, which are an adaptation to manifolds with boundary of the original ones introduced by Brendle for the study of the Yamabe flow on closed Riemannian manifolds of dimension …

Proves inequality for maximal spacelike submanifolds in Minkowski space.

Sharp inequalities for manifolds with nonnegative curvature.

In this paper, we provide some remarks on the scalar curvature rigidity theorem of Brendle and Marques in \cite{BrendleMarques}. The main result is that Brendle and Marques' theorem holds on a geodesic ball larger than that specified in [2].

In 1992, motivated by Riemann mapping theorem, Escobar considered a version of Yamabe problem on manifolds of dimension n greater than 2 with boundary. The problem consists in finding a conformal metric such that the scalar curvature is zero and the mean curvature is constant on the boundary. By using a local test func…

In a recent paper, Brendle proved that the inscribed radius of closed embedded mean convex hypersurfaces moving by mean curvature flow is at least 1/((1+δ)H) at all points with H > C(δ,M_0). In this note, we give a shorter proof of Brendle's estimate, and of a more general result for alpha-Andrews flows, based on our r…

In this short note, we show that the assumption "convex" in Theorem 7 of Brendle-Eichmair's paper \cite{BE} is unnecessary.

In this short paper, we will give a simple and transcendental proof for Mok's theorem of the generalized Frankel conjecture. This work is based on the maximum principle in \cite{BS2} proposed by Brendle and Schoen.

The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of Böhm-Wilking, that the nor…

Simon Brendle's result extended to manifolds with positive isotropic curvature of dimension at least nine.

We extend to higher dimensions earlier sharp bounds for the area of two dimensional free boundary minimal surfaces contained in a geodesic ball of the round sphere. This follows work of Brendle and Fraser-Schoen in the euclidean case.

In this note we prove that there is no constant $C$, depending on the genus of the surface, such that every element in the mapping class group can be written as a product of at most $C$ torsion elements, answering a question of T. E. Brendle and B. Farb in the negative.

We generalize Brendle's geometric inequality considered in \cite{B} to static manifolds. The inequality bounds the integral of inverse mean curvature of an embedded mean-convex hypersurface by geometric data of the horizon. As a consequence, we obtain a reverse Penrose inequality on static asymptotically locally hyperb…

Recently Brendle-Huisken introduced a fully nonlinear flow $G$. Their aim was to extend the surgery algorithm of Huisken-Sinestrari, into the Riemannian setting. The aim of this paper is to go through the details on how to perform neck detection for a closed, embedded hypersurface $M_0$ in $\mathbb{R}^{n+1}$ undergoing…

Proves spacetime positive mass theorem in all dimensions.

New inequalities derived for hyperbolic space via specific flows.

In a recent paper, Brendle showed the uniqueness of the Bryant soliton among 3-dimensional $κ$-solutions. In this paper, we present an alternative proof for this fact and show that compact $κ$-solutions are rotational symmetric. Our proof arose from independent work relating to our Strong Stability Theorem for singular…

Study proves rigidity for Heintze-Karcher inequality in substatic manifolds.

Study on rigidity with non-negative intermediate curvature on low-dimensional manifolds.

We extend the Lyapunov-Schmidt analysis of outlying stable CMC spheres in the work of S. Brendle and the second-named author to the "far-off-center" regime and to include general Schwarzschild asymptotics. We obtain sharp existence and non-existence results for large stable CMC spheres that depend very delicately on th…

Paper proves new isoperimetric inequality for minimal submanifolds with free boundary.

New proof classifies ancient flows in 3D space.

Log-Sobolev inequality proven for submanifolds in specific types of manifolds.

Researchers prove rigidity of convex polytopes in hyperbolic space using spinor techniques.

The Penrose inequality in Minkowski is a geometric inequality relating the total outer null expansion and the area of closed, connected and spacelike codimension-two surfaces S in the Minkowski spacetime, subject to an additional convexity assumption. In a recent paper, Brendle and Wang find a sufficient condition for …

Optimizes transport on submanifolds for curvature inequalities.

For any asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder we show that the link of the asymptotic cone must separate the unit sphere into exactly two connected components, both diffeomorphic to the self-shrinker. Combining this with recent work of Brendle, we conclude that the r…

Motivated by Brendle-Marques-Neves' counterexample to the Min-Oo's conjecture, we prove a volume constrained scalar curvature rigidity theorem which applies to the hemisphere.

Alexandrov immersed surfaces maintain their properties under mean curvature flow.

Optimal systolic inequality proved for manifolds with positive bi-Ricci curvature.

Paper proves optimal systolic inequality for manifolds with positive triRic curvature.

In this short note, we prove that the only simply connected noncompact three-dimensional Type I $κ$-solution to the Ricci flow is the shrinking cylinder. This work can be regarded as a generalization of Cao and Chow, and a complement of Ding and Ni. Up to this point, three-dimensional $κ$-solutions of Type I are comple…

The paper proves $L^p$-Sobolev inequalities for minimal submanifolds.

We prove that the area of a free boundary minimal surface $Σ^2 \subset B^n$, where $B^n$ is a geodesic ball contained in a round hemisphere $\mathbb{S}^n_+$, is at least as big as that of a geodesic disk with the same radius as $B^n$; equality is attained only if $Σ$ coincides with such a disk. More generally, we prove…

Generators found for a specific subgroup of braid groups.

In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as "weighted'' mean curvatures, which extend the work \cite{Mon, Brendle,BE} considering constant mean curvature functions. Secondly, …

In this paper we compute a presentation for the group of ring motions of the split union of a Hopf link with Euclidean components and a Euclidean circle. A key part of this work is the study of a short exact sequence of groups of ring motions of general ring links in $\mathbb{R}^3$. This sequence allowed us to build th…

We introduce the notion of Canonical Expanding Ricci Soliton, and use it to derive new Harnack inequalities for Ricci flow. This viewpoint also gives geometric insight into the existing Harnack inequalities of Hamilton and Brendle.

New non-perturbative counterexamples to Min-Oo's Conjecture are created.

The paper resolves a conjecture about curvature conditions on manifolds.

Logarithmic Sobolev inequality proven for non-compact self-shrinkers.

Proves planarity and convexity for ancient solutions of mean curvature flow.

Extends a result on manifolds with specific curvature properties.

The paper finds a lower bound for a Neumann eigenvalue of surfaces with flat ends.

New Sobolev inequalities found for curved spaces.

Let $(M^n,g,\nabla f)$, $n\geq 3$, be an expanding gradient Ricci soliton with nonnegative sectional curvature whose asymptotic cone is isometric to $C(\mathbb{S}^{n-1}(c))$ where $\mathbb{S}^{n-1}(c)$ is the standard $(n-1)$-sphere of curvature $1/c^2$, with $c\in(0,1)$. We prove that if the convergence to the asympto…

New classification for certain compact manifolds with positive isotropic curvature.

We say that a nonnegatively curved manifold $(M,g)$ has quarter pinched flag curvature if for any two planes which intersect in a line the ratio of their sectional curvature is bounded above by 4. We show that these manifolds have nonnegative complex sectional curvature. By combining with a theorem of Brendle and Schoe…