We prove that functions defined on a lattice in a finite dimensional torus with bounded finite differences can be smoothly extended to the whole torus, and relate the bounds on the extensions derivatives with bounds on the original functions finite differences.
We study integral and pointwise bounds on the second fundamental form of properly immersed self-shrinkers with boundedHA. As applications, we discuss gap and compactness results for self-shrinkers.
A classical theorem of Kuratowski says that every Baire one function on a G_delta subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire
one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index beta(f). We prove a refinement of Kuratowskis theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such that beta_{Y}(f)<omega^{alpha}, alpha < omega_1, then f has an extension F onto X so that beta_X(F)is not more than omega^{alpha}. We also show that if f is a continuous real valued function on Y, then f has an extension F onto X so that beta_{X}(F)is not more than 3. An example is constructed to show that this result is optimal.
In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of codimension
$geq 4$. We also establish a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case. These results are based on a compactness theorem for Ricci flows with bounded scalar curvature, which states that any sequence of such Ricci flows converges, after passing to a subsequence, to a metric space that is smooth away from a set of codimension $geq 4$. In the course of the proof, we will also establish $L^{p < 2}$-curvature bounds on time-slices of such flows.
In this paper we prove a compactness result for Ricci flows with bounded scalar curvature and entropy. It states that given any sequence of such Ricci flows, we can pass to a subsequence that converges to a metric space which is smooth away from a se
t of codimension $geq 4$. The result has two main consequences: First, it implies that singularities in Ricci flows with bounded scalar curvature have codimension $geq 4$ and, second, it establishes a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case. In the course of the proof, we will also establish the following results: $L^{p < 4}$ curvature bounds, integral bounds on the curvature radius, Gromov-Hausdorff closeness of time-slices, an $varepsilon$-regularity theorem for Ricci flows and an improved backwards pseudolocality theorem.
We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows techniques, we partially extend the
nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed manifolds with (sectional curvature) local bounded covering geometry, to manifolds with (global) Ricci boundedcovering geometry.