arXiv research

Proves upper bound on filling radius for manifolds with positive scalar curvature.

Uniform curvature bounds for regularized metrics with bounds on Ricci tensor and injectivity radius.

Upper bound for conjugate radius in open manifolds with scalar curvature and spectrum constraints.

Uniformly positive scalar curvature implies a lower bound on injectivity radius.

The paper bounds bandwidth and focal radius for manifolds with positive isotropic curvature.

Lower bound on boundary injectivity radius for specific tubes.

The paper proves estimates and theorems for Kähler manifolds.

Upper estimate of Dirac eigenvalue linked to hyperspherical radius.

The paper improves bounds on injectivity radius for manifolds with positive scalar curvature.

Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.

Study on curvature in finitely generated groups, showing positive curvature in specific cases.

The study shows a 3D manifold's macroscopic dimension is 1 under specific curvature constraints.

Study gives bounds on filling radius for Riemannian manifolds.

Study finds shortest geodesic loops on Stiefel manifold and calculates its injectivity radius.

Sharp upper bounds on inscribed radius for metric spaces with convex boundary.

Injectivity radius on Stiefel manifold is π.

Generalizes a soul-bound for noncompact Alexandrov spaces.

Sharp focal radius estimate for hypersurfaces in manifolds with positive curvature.

In this note we relate the geometric notion of fill radius with the fundamental group of the manifold. We prove: ''Suppose that a closed Riemannian manifold M satisfies the property that its universal cover has bounded fill radius. Then the fundamental group of M is virtually free.'' We explain the relevance of this th…

Lower bounds on curvature integral for manifolds with curvature constraints.

A canal surface is the envelope of a moving sphere with varying radius, defined by the trajectory C(t) (spine curve) of its center and a radius function r(t). In this paper, we investigate when parameter curves of the canal surface are also lines of curvature. Last of all, for special spine curves we obtain the radius …

Paper generalizes inscribed radius estimate to hyperbolic Einstein manifolds.

The study proves triviality and rigidity results for Ricci solitons and estimates their conjugate radius.

We construct a uniform local bound of curvature operator from local bounds of Ricci curvature and injectivity radius among all $n$-dimensional Ricci flows. Thus new compactness theorems for the Ricci flow and Ricci solitons are derived. In particular, we show that every Ricci flow with $|Ric|\leq K$ must satisfy $|Rm|\…

Recall that the radius of a compact metric space $(X, dist)$ is given by $rad\ X = \min_{x\in X} \max_{y\in X} dist(x,y)$. In this paper we generalize Berger's $\frac{1}{4}$-pinched rigidity theorem and show that a closed, simply connected, Riemannian manifold with sectional curvature $\geq 1$ and radius $\geq \fracπ{2…

We study the spectrum of complete noncompact manifolds with bounded curvature and positive injectivity radius. We give general conditions which imply that their essential spectrum has an arbitrarily large finite number of gaps. In particular, for any noncompact covering of a compact manifold, there is a metric on the b…

We consider a family of embedded, mean convex hypersurfaces which evolve by the mean curvature flow. It follows from general results of White that the inscribed radius at each point on the surface is at least $\frac{c}{H}$, where $c$ is a constant that depends only on the initial data. Andrews recently gave a new proof…

We prove that complete submanifolds, on which the Omori-Yau weak maximum principle for the Hessian holds, with low codimension and bounded by cylinders of small radius must have points rich in large positive extrinsic curvature. The lower the codimension is, the richer such points are. The smaller the radius is, the la…

We prove a sharp estimate for the inscribed radius under certain fully nonlinear curvature flows. This estimate is asymptotically sharp on cylinders.

Study proves inequality for hypersurfaces and shows almost extremals are close to Wulff shape.

We review recent work on the local geometry and optimal regularity of Lorentzian manifolds with bounded curvature. Our main results provide an estimate of the injectivity radius of an observer, and a local canonical foliations by CMC (Constant Mean Curvature) hypersurfaces, together with spatially harmonic coordinates.…

The paper extends radius estimates for stable hypersurfaces in 2, 3, and 4 dimensions.

We give a new and complete proof of Hamilton's injectivity radius estimate for sequences with bounded and almost nonnegative curvature operators, unbounded diameters, and bump-like origins. Such sequences arise in particular from dilations about a singularity of the Ricci flow on a 3-manifold.

In this paper we obtain a simple upper bound for the infimum of the Ricci curvatures of a complete Riemannian manifold with nonzero injectivity radius i(M) depending only on of the i(M). In case of rigidity the Riemannian manifold must be an Euclidean sphere(Euclidean space) conform the injectivity radius be finite(inf…

In this paper, we study the injectivity radius bound for 3-d Ricci flow. As applications we show the long time existence of the Ricci flow with positive Ricci curvature. We also partially settle a question in page 302 of the book of Chow-Lu-Ni (2006).

In this article we pose the problem of existence and uniqueness of convex body for which the projection curvature radius function coincides with given function. We find a necessary and sufficient condition that ensures a positive answer to both questions and suggest an algorithm of construction of the body. Also we fin…

First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space under certain condition for the radius function. Next, we prove that the …

Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,α}$ convergence techniques, we prove several differentiable stability and sphere theorem versions of these results and apply those also to the study of Einstein m…

A method to fix radius distortion in generative models on curved spaces.

We give new estimates for the extrinsic radius of compact hypersurfaces of the Euclidean space and the open hemisphere in terms of high order mean curvatures. Then we prove pinching results corresponding to theses estimates. We show that under a suitable pinching condition, the hypersurface is diffeomorphic and almost …

We give an optimal estimate for the norm of any submanifold's second fundamental form in terms of its focal radius and the lower sectional curvature bound of the ambient manifold. This is a special case of a similar theorem for intermediate Ricci curvature, and leads to a $C^{1,α}$ compactness result for submanifolds, …

Uniform eigenvalue bounds for Hodge Laplacian on manifolds with Ricci curvature and injectivity radius bounds.

The paper proves bounds on curvature and injectivity radius for convex sums of Riemannian metrics.

We show that the focal radius of any submanifold $N$ of positive dimension in a manifold $M$ with sectional curvature greater than or equal to $1$ does not exceed $\frac{π}{2}.$ In the case of equality, we show that $N$ is totally geodesic in $M$ and the universal cover of $M$ is isometric to a sphere or a projective s…

The study proves a rigidity theorem for compact manifolds with boundary.

In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$, $\operatorname{conv}(p)\ge \min\{\frac{1}{2}\operatorname{inj}(p),\operatorname{foc}(B_{\operatorname…

Estimates the dual Thurston norm for foliations on negative curvature 3-manifolds.

In a recent paper, Brendle proved that the inscribed radius of closed embedded mean convex hypersurfaces moving by mean curvature flow is at least 1/((1+δ)H) at all points with H > C(δ,M_0). In this note, we give a shorter proof of Brendle's estimate, and of a more general result for alpha-Andrews flows, based on our r…