arXiv research

The paper resolves a conjecture about curvature conditions on manifolds.

Study shows expanding Ricci solitons from specific metric cones.

In this article we use Ricci flow to show that complete PIC1 manifolds with maximal volume growth are diffeomorphic to $\mathbb{R}^n$. One of the key ingredients is local estimates of curvature lower bounds on an initial time interval of the Ricci flow. As another application of these estimates we obtain pseudolocality…

Surveying problems with positive curvature forms, focusing on Ricci flow.

This paper extends 3D results to higher dimensions, proving compactness for PIC1 pinched manifolds.

We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short). In the first part of this paper, we prove new curvature pinching estimates which ensure that blow-up limits are uniformly PIC in all dimensions. Moreover, in dimension $n \geq 12$, we show that blow-up limits are wea…

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 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 …

New curvature condition proves rigidity of Bryant Ricci solitons.

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 paper we prove classification results for gradient shrinking Ricci solitons under two invariant conditions, namely nonnegative orthogonal bisectional curvature and weakly PIC1, without any curvature bound. New results on ancient solutions for the Ricci and Kähler-Ricci flow are also obtained. The main new featu…

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

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

Proves inequality for maximal spacelike submanifolds in Minkowski space.

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…

New inequalities derived for hyperbolic space via specific flows.

Sharp inequalities for manifolds with nonnegative curvature.

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

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

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…

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.

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…

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

In this paper, we construct a pyramid Ricci flow starting with a complete Riemannian manifold $(M^n,g_0)$ that is PIC1, or more generally satisfies a lower curvature bound $K_{IC_1}\geq -α_0$. That is, instead of constructing a flow on $M\times [0,T]$, we construct it on a subset of space-time that is a union of parabo…

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.

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.

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

The paper studies Ricci flow on manifolds with boundary, proving existence, uniqueness, and boundary conditions preservation.

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

Extends a result on manifolds with specific curvature properties.

Proves pinched Ricci curvature conjecture in all dimensions.

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.

New Sobolev inequalities found for curved spaces.

Proves spacetime positive mass theorem in all dimensions.

Study Ricci flows on manifolds, proving they behave like self-similar solutions and confirming a conjecture.

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

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…

It is proved by Brendle in [4] that the equatorial disk $D^k$ has least area among $k$-dimensional free boundary minimal surfaces in the Euclidean ball $B^n$. By comparing the excess of free boundary minimal surfaces with the excess of the associated cones over the boundary, we prove the existence of a gap for the area…

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…

In this note, we study the curvature flow to Nirenberg problem on $S^2$ with non-negative nonlinearity. This flow was introduced by Brendle and Struwe. Our result is that the Nirenberg problems has a solution provided the prescribed non-negative Gaussian curvature $f$ has its positive part, which possesses non-degenera…

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 …

We consider closed orientable hypersurfaces in a wide class of warped product manifolds, which include space forms, deSitter-Schwarzschild and Reissner-Nordström manifolds. By using a new integral formula or Brendle's Heintze-Karcher type inequality, we present some new characterizations of umbilic hypersurfaces. These…

Alexandrov's theorem asserts that spheres are the only closed embedded constant mean curvature hypersurfaces in space forms. In this paper, we consider Alexandrov's theorem in warped product manifolds and prove a rigidity result in the spirit of Alexandrov's theorem. Our approach generalizes the proofs of Reilly and Ro…

Proves inequality for tensor fields on curved spaces.

New proof classifies ancient flows in 3D space.

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