Embeddedness for some nonconvex curves during the area-presering and length-preserving curve shortening flow (part I)

Abstract: 
In this talk we consider families of embedded curves in the plane which evolve by the area-presering or the length-preserving curve shortening flow (apcsf and lpcsf). We show preservation of embeddedness during the flow if the angels between any two tangent vectors of the initial curve lie between -pi and 3pi (plus additional restriction for the lpcsf).

Geometric singularities and a flow tangential to the Ricci flow

Abstract:
In 2012, Gigli and Mantegazza introduced a new geometric flow via heat kernels. They demonstrated that this flow is tangential to the Ricci flow in a suitable weak sense for smooth, compact Riemannian manifolds. A salient feature of this flow is that it can be given meaning for compact RCD metric spaces by interpreting the equation distributionally as a flow of the distance metric. Gigli and Mantegazza further show that the two formulations agree for the smooth, compact manifold case. As a consequence, this flow can be successfully defined for spaces containing certain singularities. An important question is to understand regularity - do singularities disappear along the flow, or are they retained? The quintessential example has been to study manifolds with conical singularities. In our work, we partially address this regularity question by studying spaces with "geometric singularities", by which we mean a smooth manifold but with a non-smooth metric. When such spaces are also RCD metric spaces with singularities on a closed subset, we obtain a metric tensor on the open non-singular part with regularity corresponding to the regularity of the initial heat kernel. In particular, we demonstrate that a manifold with a finite number of geometric conical singularities remains a smooth manifold away from the cone points for all time along the flow. For "rough" initial metrics, where we expect only continuity of the flow, we demonstrate connections between regularity of the flow and homogeneous Kato square root estimates.

Stable capillary hypersurfaces in a wedge

Let S be an immersed stable hypersurface of constant mean curvature in a wedge bounded by two hyperplanes in R^n . Suppose that S meets those two hyperplanes in constant contact angles and is disjoint from the edge of the wedge. We will show that if the boundary of S is embedded for n = 3, or convex for n >= 4, then S is part of the sphere. This is a joint work with M. Koiso.

Sharp estimates for $(m+1)$-convex mean curvature flows
In a major recent breakthrough, Andrews discovered a simple proof of the quantitative $\kappa$-non-collapsing property of embedded mean convex mean curvature flows first observed by Sheng and Wang. This estimate states that the pointwise quantity obtained by multiplying the radius of the largest inscribed ball with the mean curvature does not deteriorate under the flow. One motivation for considering such a quantity is that it degenerates on certain `collapsed' singularity models such as Grim hyperplanes (which appear in immersed flows) and planes of multiplicity two; thus, in particular, non-collapsing rules out `bad' singularities. The beauty of Andrews' argument is that it requires nothing more than the maximum principle, applied to a certain extrinsic quantity related to the inscribed curvature (the inverse of the inscribed radius). Moreover, the argument applies also to the exscribed curvature, yielding a stronger `two-sided' non-collapsing property. We note that two-sided non-collapsing played a central role in the theory of mean convex mean curvature flow subsequently developed by Haslhofer and Kleiner. Recently, Brendle applied a Stampacchia iteration argument (similar to those employed by Huisken and Huisken--Sinestrari to obtain estimates for the second fundamental form, but involving several new ideas) to Andrews' geometric quantity, and thereby proved that the ratios of exscribed and inscribed curvature to the mean curvature improve along a mean convex mean curvature flow, becoming sharp at the onset of a singularity. These estimates played an important role in Brendle and Huisken's development of mean curvature flow with surgery for embedded surfaces. In our talk, we will describe the main ingredients in the arguments of Andrews and Brendle, and show how their ideas can be extended to obtain sharper estimates when the initial data satisfy stronger convexity assumptions (namely, `$(m+1)$-convexity' for any $m\in\{0,\dots,n-1\}$). Along the way, we will describe a new `$m$-convexity' estimate for the mean curvature flow which may be of independent interest. Time permitting, we will describe how the methods can be adapted to obtain analogous estimates for flows by more general functions of curvature satisfying standard homogeneity and concavity requirements. Cheers, Mat