arXiv subjects
Publications and source records attributed to Or Hershkovits.
We prove that the moduli space of 2-convex embedded n-spheres in R^{n+1} is path-connected for every n. Our proof uses mean curvature flow with surgery and can be seen as an extrinsic analog to Marques' influential proof of the path-connectedness of the moduli space of positive scalar curvature metics on three-manifolds.
In this short article we investigate the topology of the moduli space of two-convex embedded tori $S^{n-1}\times S^1\subset \mathbb{R}^{n+1}$. We prove that for $n \geq 3$ this moduli space is path-connected, and that for $n = 2$ the connected components of the moduli space are in bijective correspondence with the knot classes associated to the embeddings. Our proof uses a variant of mean curvature flow with surgery developed in our earlier article (arXiv:1607.05604) where neck regions are deformed to tiny strings instead of being cut out completely, an approach which preserves the global topology, embeddedness, as well as two-convexity.
In this paper we discuss a simple relation, which was previously missed, between the high co-dimensional isoperimetric problem of finding a filling with small volume to a given cycle, and extinction estimates for singular, high co-dimensional, mean curvature flow. The utility of this viewpoint is first exemplified by two results which, once casted in the light of this relation, are almost self evident. The first is a genuine, 5-lines proof, for the isoperimetric inequality for $k$-cycles in $\mathbb{R}^n$, with a constant differing from the optimal constant by a factor of only $\sqrt{k}$, as opposed to a factor of $k^k$ produced by all of the other soft methods (like Michael-Simon's or Gromov's). The second is a 3-lines proof of a lower bound for extinction for arbitrary co-dimensional, singular, mean curvature flows starting from cycles, generalizing the main result of a paper of Giga and Yama-uchi. We then turn to use the above mentioned relation to prove a bound on the parabolic Hausdorff measure of the space time track of high co-dimensional, singular, mean curvature flow starting from a cycle, in terms of the mass of that cycle. This bound is also reminiscent of a Michael-Simon Isoperimetric inequality. To prove it, we are lead to study the geometric measure theory of Euclidean rectifiable sets in parabolic space, and prove a co-area formula in that setting. This formula, the proof of which occupies most this paper, may be of independent interest
We study the existence and uniqueness of smooth mean curvature flow, in arbitrary dimension and co-dimension, emanating from so called $k$-dimensional $(\varepsilon,R)$ Reifenberg flat sets in $\mathbb{R}^n$. Our results generalize the ones from a previous paper by the author, in which the co-dimension one case (i.e. $k=n-1$) was studied. For $\varepsilon$ fixed, this class is general enough to include (i) all $C^2$ sub-manifolds (ii) all Lipschitz sub-manifolds with Lipschitz constant less than $\varepsilon$ (iii) some sets with Hausdorff dimension larger than $k$. The Reifenberg condition, roughly speaking, says that the set has a weak metric notion of a $k$-dimensional tangent plane at every point and scale, but those tangents are allowed to tilt as the scales vary. We show that if the Reifenberg parameter $\varepsilon$ is small enough, the (arbitrary co-dimensional) level set flow (in the sense of Ambrosio-Soner ) is non fattening, smooth and attains the initial value in the Hausdorff sense. In particular, our result generalizes a result of Wang and, in fact, all known existence and uniqueness results for smooth mean curvature flow in arbitrary co-dimension. The largest deviation from the proof of the co-dimension case comes in the proof of uniqueness (i.e. non-fattening), where one is forced to work with the viscosity notion of the high co-dimensional level set flow, rather than Ilmanen's more geometric definition. This study leads to a general (short time) smooth uniqueness result, generalizing the one for evolution of smooth sub-manifolds, which may be of independent interest, even in co-dimension one.
We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove a stumbling block that has been left after the work of White and Haslhofer-Kleiner, and thus allow us to extend the structure theory for mean convex level set flow to general ambient manifolds of arbitrary dimension.
In this paper, we prove short time existence and uniqueness of smooth evolution by mean curvature in $\mathbb{R}^{n+1}$ starting from any $n$-dimensional $(\varepsilon,R)$-Reifenberg flat set with $\varepsilon$ sufficiently small. More precisely, we show that the level set flow in such a situation is non-fattening and smooth. These sets have a weak metric notion of tangent planes at every small scale, but the tangents are allowed to tilt as the scales vary. As this class is wide enough to include some fractal sets, this provides the first example (when $n>1$) of unique smoothing by mean curvature flow of sets with Hausdorff dimension $> n$.
In this short article, we prove the existence of ancient solutions of the mean curvature flow that for t -> 0 collapse to a round point, but for t -> -infinity become more and more oval: near the center they have asymptotic shrinkers modeled on round cylinders S^j x R^n-j and near the tips they have asymptotic translators modeled on Bowl^j+1 x R^n-j-1. We also give a characterization of the round shrinking sphere among ancient alpha-Andrews flows. Our proofs are based on the recent estimates of Haslhofer-Kleiner.