Related papers: On Area Growth in Sol

In this paper, we prove uniform lower bounds on the volume growth of balls in the universal covers of Riemannian surfaces and graphs. More precisely, there exists a constant $\delta>0$ such that if $(M,hyp)$ is a closed hyperbolic surface…

We examine the space of surfaces in $\RR^{3}$ which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the space $\Mk$ of…

This study focuses on a radial alignment between Parker Solar Probe (PSP) and Solar Orbiter (SolO) on the 29$^{\text{th}}$ of April 2021 (during a solar minimum), when the two spacecraft were respectively located at $\sim 0.075$ and $\sim…

We consider the Lie group $SU_2$ endowed with a left-invariant axisymmetric Riemannian metric. This means that a metric has eigenvalues $I_1 = I_2, I_3 > 0$. We give an explicit formula for the diameter of such metric. Other words, we…

We address the problem of analyticity up to the boundary of solutions to the Euler equations in the half space. We characterize the rate of decay of the real-analyticity radius of the solution $u(t)$ in terms of $\exp{\int_{0}^{t} \Vert…

Consider a closed manifold $M$ with two Riemannian metrics: one hyperbolic metric, and one other metric $g$. What hypotheses on $g$ guarantee that for a given radius $r$, there are balls of radius $r$ in the universal cover of $(M, g)$ with…

In this paper we study the Weil-Petersson geometry of $\overline{\mathcal{M}_{g,n}}$, the compactified moduli space of Riemann surfaces with genus g and n marked points. The main goal of this paper is to understand the growth of the…

Working in positive characteristic, we show how one can use information about the dimension of moduli spaces of rational curves on a Fano variety $X$ over $\mathbb{F}_q$ to obtain strong estimates for the number of $\mathbb{F}_q(t)$-points…

We prove that the boundary of an orbit space or more generally a leaf space of a singular Riemannian foliation is an Alexandrov space in its intrinsic metric, and that its lower curvature bound is that of the leaf space. A rigidity theorem…

Main Theorem: Spaces of r-branch point 3-cycle covers, degree n or Galois of degree n!/2 have one (resp. two) component(s) if r=n-1 (resp. r\ge n). Improves Fried-Serre on deciding when sphere covers with odd-order branching lift to…

Cone spherical surfaces are orientable Riemannian surfaces with constant curvature one and a finite set of conical singularities. A subset of these surfaces, referred to as dihedral surfaces, is characterized by their monodromy groups,…

If the n-dimensional unit sphere is covered by finitely many spherically convex bodies, then the sum of the inradii of these bodies is at least {\pi}. This bound is sharp, and the equality case is characterized.

In a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inequality. Namely, as discussed in the paper, its area must be bounded above by $4\pi/c$,…

In this paper, we prove the strong Morse inequalities for the area functional in the space of embedded tori and spheres in the three sphere. As a consequence, we prove that in the three dimensional sphere with positive Ricci curvature,…

Let's have $n$ points in the space such that the maximum distance between any of them is $a$. We prove that there exists a sphere of radius $r \leq a \frac{\sqrt(6)}{4}$ that contains in its interior or on its surface all these points.…

In Ahlfors' covering surface theory, it is well known that there exists a positive constant $h$ such that for any nonconstant holomorphic mapping $f:% \bar{\Delta}\to S,$ if $f(\Delta)\cap \{0,1,\infty \}=\emptyset ,$ then% A(f,\Delta)\leq…

Given a lattice polygon $P$ with $g$ interior lattice points, we associate to it the moduli space of tropical curves of genus $g$ with Newton polygon $P$. We completely classify the possible dimensions such a moduli space can have. For…

Let X be a smooth algebraic variety on which a solvable Lie group acts freely on a dense open orbit. Such varieties include generalized flag varieties, toric varieties, Bott-Samelson varieties, and many spherical varieties, as well as…

In this paper we generalize a result in [1], showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of…

We show that the volume of any Riemannian metric on a three sphere is bounded below by the length of the shortest closed curve that links its antipodal image. In particular, the volume is bounded below by the minimum of the length of the…