Related papers: A note on the compactness theorem for 4d Ricci shr…

We prove a local gap theorem for Ricci shrinkers, which states that if the local $\mu$-functional at scale $1$ on a large ball centered at the minimum point of the potential function is close enough to $0$, then the shrinker must be the…

A compactness theorem for volume-constrained almost-critical points of elliptic integrands is proven. The result is new even for the area functional, as almost-criticality is measured in an integral rather than in a uniform sense. Two main…

In this paper we prove weak L^{1,p} (and thus C^{\alpha}) compactness for the class of uniformly mean-convex Riemannian n-manifolds with boundary satisfying bounds on curvature quantities, diameter, and (n-1)-volume of the boundary. We…

We characterize group compactifications of discrete groups for which there exists an equivariant retraction onto the boundary. In particular, we prove an equivariant analogue of Brouwer's No-Retraction theorem for large classes of group…

For complete shrinking (gradient) Ricci solitons, we observe a quantification of Wylie's result that the fundamental group is finite.

In Theorem 3.1 of [12], we proved a rigidity result for self-shrinkers under the integral condition on the norm of the second fundamental form. In this paper, we relax the such bound to any finite constant (see Theorem 4.4 for details).

For compact submanifolds in Euclidean and Spherical space forms with Ricci curvature bounded below by a function $\alpha(n,k,H,c)$ of mean curvature, we prove that the submanifold is either isometric to the Einstein Clifford torus, or a…

In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume…

For $n$-dimensional Riemannian manifolds $M$ with Ricci curvature bounded below by $-(n-1)$, the volume entropy is bounded above by $n-1$. If $M$ is compact, it is known that the equality holds if and only if $M$ is hyperbolic. We extend…

Define $r_4(N)$ to be the largest cardinality of a set $A \subset \{1,\dots,N\}$ which does not contain four elements in arithmetic progression. In 1998 Gowers proved that \[ r_4(N) \ll N(\log \log N)^{-c}\] for some absolute constant…

Motivated by the recently proposed bounds on the slow-roll parameters for scalar potentials arising from string/M-theory compactifications, a.k.a. the Refined de Sitter Swampland conjecture, we explore the sharpness of such constraints…

We prove some boundary rigidity results for the hemisphere under a lower bound for Ricci curvature. The main result can be viewed as the Ricci version of a conjecture of Min-Oo.

We consider the 3D smectic energy $$\mathcal{E}_{\epsilon }\left( u\right) =\frac{1}{2}\int_{\Omega }\frac{1}{\varepsilon } \left( u_z-\frac{( u_x)^{2}+( u_y)^{2}}{2}\right) ^{2}+\varepsilon \left( u_{xx}+u_{yy}\right)^{2}\,dx\,dy\,dz. $$…

For Riemannian manifolds with a smooth measure $(M, g, e^{-f}dv_{g})$, we prove a generalized Myers compactness theorem when Bakry--Emery Ricci tensor is bounded from below and $f$ is bounded.

In this paper, we first prove a compactness theorem for the space of closed embedded $f$-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry-\'{E}mery Ricci curvature. Then we give a Lichnerowicz type lower…

Let $B_2(p)$ be an $n$-dimensional smooth geodesic ball with Ricci curvature $\geq-(n-1)\kappa^2$ for some $\kappa\geq0$. We establish the Sobolev inequality and the uniform Neumann-Poincar\'e inequality on each minimal graph over $B_1(p)$…

In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci solitons with nonnegative Ricci curvature (outside a compact set $K$). More precisely, we prove that the norm of the curvature tensor $Rm$ and…

We prove sufficient and necessary conditions for compactness of Calder\'on-Zygmund operators on the endpoint from $L^{\infty }(\mathbb R)$ into ${\rm CMO}(\mathbb R)$. We use this result to prove compactness on $L^{p}(\mathbb R)$ with…

We prove a lower bound estimate for the first non-zero eigenvalue of the Witten-Laplacian on compact Riemannian manifolds. As an application, we derive a lower bound estimate for the diameter of compact gradient shrinking Ricci solitons.…

We consider the volume-normalized Ricci flow close to compact shrinking Ricci solitons. We show that if a compact Ricci soliton $(M,g)$ is a local maximum of Perelman's shrinker entropy, any normalized Ricci flow starting close to it exists…