Related papers: Compactness and rigidity of self-shrinking surface…

Let $(\Sigma, g_1)$ be a compact Riemann surface with conical singularites of angles in $(0, 2\pi)$, and $f: \Sigma\to\mathbb R$ be a positive smooth function. In this paper, by establishing a sharp quantization result, we prove the…

We consider immersions admitting uniform representations as an L-Lipschitz graph. In codimension 1, we show compactness for such immersions for arbitrary fixed finite L and uniformly bounded volume. The same result is shown in arbitrary…

We prove that if the initial hypersurface of the mean curvature flow in spheres satisfies a sharp pinching condition, then the solution of the flow converges to a round point or a totally geodesic sphere. Our result improves the famous…

In this paper, We define a $\mathcal{F}$-functional and study $\mathcal{F}$-stability of $\lambda$-hypersurfaces, which extend a result of Colding-Minicozzi. Lower bound growth and upper bound growth of area for complete and non-compact…

We consider surfaces which minimize a nonlocal perimeter functional and we discuss their interior regularity and rigidity properties, in a quantitative and qualitative way, and their (perhaps rather surprising) boundary behavior. We present…

The present work constitutes the third installment in a series of investigations devoted to discrete conformal structures on surfaces with boundary. In our preceding works \cite{X-Z DCS1, X-Z DCS2}, we established, respectively, a…

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find pinching conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions…

We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a $C^{1,\lambda}$-a-priori bound for surfaces for which this functional is finite. In fact, it turns out…

We prove existence for many examples of shrinkers by producing compact, smoothly embedded surfaces that, under mean curvature flow, develop singularities at which the shrinkers occur as blowups.

Let M be a 3-manifold (possibly with boundary). We show that, for any positive integer g, there exists an open nonempty set of metrics on M for each of which there are stable compact embedded minimal surfaces of genus g with arbitrarily…

We obtain a compactness result for Fano manifolds and K\"ahler Ricci flows. Comparing to the more general Riemannian versions by Anderson and Hamilton, in this Fano case, the curvature assumption is much weaker and is preserved by the…

In this paper, we introduce a definition of $\lambda$-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that $\lambda$-hypersurfaces are critical points of the weighted area functional for the…

In [J. Cantarella, J. Parsley, J. Geom. Phys. 60:1127 (2010)] Cantarella and Parsley introduced the notion of submanifold helicity. In the present paper we investigate properties of surface helicity and in particular answer two open…

We consider a smooth closed surface $M$ of fixed genus $\geqslant 2$ with a Riemannian metric $g$ of negative curvature with fixed total area. The second author has shown that the topological entropy of geodesic flow for $g$ is greater than…

We obtain improved local well-posedness results for the Lorentzian timelike minimal surface equation. In dimension $d=3$, for a surface of arbitrary co-dimension, we show a gain of $1/3$ derivative regularity compared to a generic equation…

This paper studies singularities of mean curvature flows with integral mean curvature bounds $H \in L^\infty L^p_{loc}$ for some $p \in ( n, \infty]$. For such flows, any tangent flow is given by the flow of a stationary cone $\mathbf{C}$.…

We prove a smooth compactness theorem for the space of elasticae, unless the limit curve is a straight segment. As an application, we obtain smooth stability results for minimizers with respect to clamped boundary data.

Relative motion between soft wet solids arises in a number of applications in natural and artificial settings, and invariably couples elastic deformation and fluid flow. We explore this in a minimal setting by considering a fluid-immersed…

We localize the entropy functionals of G. Perelman and generalize his no-local-collapsing theorem and pseudo-locality theorem. Our generalization is technically inspired by further development of Li-Yau estimates along the Ricci flow. It…

In this paper we prove two theorems. The first one is a structure result that describes the extrinsic geometry of an embedded surface with constant mean curvature (possibly zero) in a homogeneously regular Riemannian three-manifold, in any…