Rank gradient of small covers

We prove that if $M \longrightarrow P$ is a small cover of a compact right-angled hyperbolic polyhedron $P$ then $M$ admits a cofinal tower of finite sheeted covers with positive rank gradient. As a corollary, if $\pi_1(M)$ is commensurable with the reflection group of $P$, then $M$ admits a cofinal tower of finite sheeted covers with positive rank gradient.Comment: accepted in Pacific Journal of Mathematic

Rank gradient in co-final towers of certain Kleinian groups

We prove that if the fundamental group of an orientable finite volume hyperbolic 3-manifold has finite index in the reflection group of a right-angled ideal polyhedra in $\mathbb{H}^3$ then it has a co-final tower of finite sheeted covers with positive rank gradient. The manifolds we provide are also known to have co-final towers of covers with zero rank gradient.Comment: 15 pages, 2 figure

A sharp inequality for Sobolev functions

Let $N\geq 5$, $a>0$, $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, $2^*=\frac{2N}{N-2}$, $2^\#=\frac{2(N-1)}{N-2}$ and $||u||^2=|\nabla u|_{2}^2+a|u|_{2}^2$. We prove there exists an $\alpha_{0}>0$ such that, for all $u\in H^1(\Omega)\setminus\{0\}$, $$\frac{S}{2^{\frac 2N}}\leq\frac{||u||^2}{|u|_{2^*}^2}\left(1+\alpha_{0}\frac{|u|_{2^\#}^{2^\#}}{||u||\cdot|u|_{2^*}^{2^*/2}}\right).$$ This inequality implies Cherrier's inequality.Comment: 4 page

A family of sharp inequalities for Sobolev functions

Let $N\geq 5$, $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, ${2^*}=\frac{2N}{N-2}$, $a>0$, $S=\inf\left\{\left. \int_{\mathbb{R}^{N}}|\nabla u|^2\,\right|\,u\in L^{2^*}(\mathbb{R}^{N}), \nabla u\in L^2(\mathbb{R}^{N}), \int_{\mathbb{R}^{N}}|u|^{2^*}=1 \right\}$ and $||u||^2=|\nabla u|_{2}^2+a|u|_{2}^2$. We define ${2^\flat}= \frac{2N}{N-1}$, ${2^\#}=\frac{2(N-1)}{N-2}$ and consider $q$ such that ${2^\flat}\leq q\leq{2^\#}$. We also define $s=2-N+\frac{q}{{2^*}-q}$ and $t=\frac{2}{N-2}\cdot \frac{1}{{2^*}-q}$. We prove that there exists an $\alpha_{0}(q,a,\Omega)>0$ such that, for all $u\in H^1(\Omega)\setminus\{0\}$, $$\frac{S}{2^{\frac 2N}}{|u|_{{2^*}}^2}\leq||u||^2+\alpha_{0} \left(\frac{||u||}{|u|_{{2^*}}^{2^*/2}}\right)^s|u|_{q}^{qt},\qquad{(I)_{q}}$$ where the norms are over $\Omega$. Inequality $(I)_{{2^\flat}}$ is due to M. Zhu.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1407.623

Bifurcation curves of a logistic equation when the linear growth rate crosses a second eigenvalue

We construct the global bifurcation curves, solutions versus level of harvesting, for the steady states of a diffusive logistic equation on a bounded domain, under Dirichlet boundary conditions and other appropriate hypotheses, when $a$, the linear growth rate of the population, is below $\lambda_2+\delta$. Here $\lambda_2$ is the second eigenvalue of the Dirichlet Laplacian on the domain and $\delta>0$. Such curves have been obtained before, but only for $a$ in a right neighborhood of the first eigenvalue. Our analysis provides the exact number of solutions of the equation for $a\leq\lambda_2$ and new information on the number of solutions for $a>\lambda_2$.Comment: This is an extended version of the published pape

Convergence of a crystalline algorithm for the motion of a simple closed convex curve by weighted curvature

Motion by weighted mean curvature is a geometric evolution law for surfaces and represents steepest descent with respect to anisotropic surface energy. It has been proposed that this motion could be computed numerically by using a "crystalline" approximation to the surface energy in the evolution law. In this paper we prove the convergence of this numerical method for the case of simple closed convex curves in the plane.Comment: 14 pages, 3 figure

Large induced subgraphs with kk vertices of almost maximum degree

In this note we prove that for every integer $k$, there exist constants $g_{1}(k)$ and $g_{2}(k)$ such that the following holds. If $G$ is a graph on $n$ vertices with maximum degree $\Delta$ then it contains an induced subgraph $H$ on at least $n - g_{1}(k)\sqrt{\Delta}$ vertices, such that $H$ has $k$ vertices of the same degree of order at least $\Delta(H)-g_{2}(k)$. This solves a conjecture of Caro and Yuster up to the constant $g_{2}(k)$.Comment: 6 page

An improved upper bound on the maximum degree of terminal-pairable complete graphs

A graph $G$ is terminal-pairable with respect to a demand multigraph $D$ on the same vertex set as $G$, if there exists edge-disjoint paths joining the end vertices of every demand edge of $D$. In this short note, we improve the upper bound on the largest $\Delta(n)$ with the property that the complete graph on $n$ vertices is terminal-pairable with respect to any demand multigraph of maximum degree at most $\Delta(n)$. This disproves a conjecture originally stated by Csaba, Faudree, Gy\'arf\'as, Lehel and Schelp.Comment: 4 page

Weighted geometric inequalities for hypersurfaces in sub-static manifolds

We prove two weighted geometric inequalities that hold for strictly mean convex and star-shaped hypersurfaces in Euclidean space. The first one involves the weighted area and the area of the hypersurface and also the volume of the region enclosed by the hypersurface. The second one involves the total weighted mean curvature and the area of the hypersurface. Versions of the first inequality for the sphere and for the adS-Reissner-Nordstr\"om manifold are proven. We end with an example of a convex surface for which the ratio between the polar moment of inertia and the square of the area is less than that of the round sphere

A spinorial approach to constant scalar curvature hypersurfaces in pseudo-hyperbolic manifolds

Using spinorial techniques, we prove, for a class of pseudo-hyperbolic ambient manifolds, a Heintze-Karcher type inequality. We then use this inequality to show an Alexandrov type theorem in such spaces