Related papers: Upper and lower bounds on the filling radius

In this note we provide several lower bounds for the volume of a geodesic ball within the injectivity radius in a $3$-dimensional Riemannian manifold assuming only upper bounds for the Ricci 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…

We will construct surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we…

We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder $B(r)\times\R^{\ell}$ in a product Riemannian manifold $N^{n-\ell}\times\R^{\ell}$. It follows that a complete hypersurface of given constant…

For three dimensional complete Riemannian manifolds with scalar curvature no less than one, we obtain the sharp upper bound of complete stable minimal surfaces' diameter.

In [16], we established Trudinger-Moser inequalities for complete noncompact Riemannian manifold on which the Ricci curvature has lower bound and the injectivity radius is strictly positive. In this note, we improve those inequalties when…

In this paper we obtain lower bound estimates of the spectrum of Laplace-Beltrami operator on complete submanifolds with bounded mean curvature, whose ambient space admits a Riemannian submersion over a Riemannian manifold with negative…

We establish a relationship between a certain notion of covering complexity of a Riemannian spin manifold and positive lower bounds on its scalar curvature. This makes use of a pairing between quantitative operator $K$-theory and Lipschitz…

Given a constant C and a smooth closed $(n-1)$-dimensional Riemannian manifold $(\Sigma, g)$ equipped with a positive function $H$, a natural question to ask is whether this manifold can be realised as the boundary of a smooth…

We investigate the compact submanifolds in Riemannian space forms of nonnegative sectional curvature that satisfy a lower bound on the Ricci curvature, that bound depending solely on the length of the mean curvature vector of the immersion.…

We consider modified scalar curvature functions for Riemannian manifolds equipped with smooth measures. Given a Riemannian submersion whose fiber transport is measure-preserving up to constants, we show that the modified scalar curvature of…

In this paper we generalize the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimizing closed hypersurface $\Sigma$ of a Riemannian 5-manifold $M$…

We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In…

Consider a sequence of closed, orientable surfaces of fixed genus $g$ in a Riemannian manifold $M$ with uniform upper bounds on mean curvature and area. We show that on passing to a subsequence and choosing appropriate parametrisations, the…

We exhibit conjugate points on the Stiefel manifold endowed with any member of the family of Riemannian metrics introduced by H\"uper et al. (2021). This family contains the well-known canonical and Euclidean metrics. An upper bound on the…

Given a complete isometric immersion $\phi: P^m \longrightarrow N^n$ in an ambient Riemannian manifold $N^n$ with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially…

In this note, we extend diameter bounds of Simon, Topping, and Wu--Zheng to submanifolds with boundary and (potentially non-compact) ambient manifolds with minor curvature restrictions. The bound is dependent on both an integral of mean…

Given a compact Riemannian manifold with boundary, we prove that the limit of a sequence of embedded, almost properly embedded free boundary minimal hypersurfaces, with uniform area and Morse index upper bound, always inherits a non-trivial…

We establish a uniform estimate for the injectivity radius of the past null cone of a point in a general Lorentzian manifold foliated by spacelike hypersurfaces and satisfying an upper curvature bound. Precisely, our main assumptions are,…

We give a simple upper bound for the upper box dimension of a backward invariant set of a $C^{1}$-diffeomorphism of a Riemannian manifold. We also estimate an upper bound for the box dimension of a forward invariant set of a $C^{1}$-mapping…