arXiv subjects

Andrea Mondino

Publications and source records attributed to Andrea Mondino.

87 records · Page 5Linked to original sources

arXiv2013

On the Bakry-Émery condition, the gradient estimates and the Local-to-Global property of RCD*(K,N) metric measure spaces

We prove higher summability and regularity of Γ(f) for functions f in spaces satisfying the Bakry-Émery condition BE(K,\infty). As a byproduct, we obtain various equivalent weak formulations of BE(K,N) and we prove the Local-to-Global property of the RCD*(K,N) condition in locally compact metric measure spaces (X,d,m), without assuming a priori the non-branching condition on the metric space.

math.AP↗

arXiv2013

A frame energy for immersed tori and applications to regular homotopy classes

The paper is devoted to study the Dirichelet energy of moving frames on 2-dimensional tori immersed in the euclidean $3\leq m$-dimensional space. This functional, called Frame energy, is naturally linked to the Willmore energy of the immersion and on the conformal structure of the abstract underlying surface. As first result, a Willmore-conjecture type lower bound is established : namely for every torus immersed in $\R^m$, $m\geq 3$, and any moving frame on it, the frame energy is at least $2π^2$ and equalty holds if and only if $m\geq 4$, the immersion is the standard Clifford torus (up to rotations and dilations), and the frame is the flat one. Smootheness of the critical points of the frame energy is proved after the discovery of hidden conservation laws and, as application, the minimization of the Frame energy in regular homotopy classes of immersed tori in $\R^3$ is performed.

math.DG↗

arXiv2013

Li-Yau and Harnack type inequalities in $RCD^*(K,N)$ metric measure spaces

Metric measure spaces satisfying the reduced curvature-dimension condition $CD^*(K,N)$ and where the heat flow is linear are called $RCD^*(K,N)$-spaces. This class of non smooth spaces contains Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded below by $K$ and dimension bounded above by $N$. We prove that in $RCD^*(K,N)$-spaces the following properties of the heat flow hold true: a Li-Yau type inequality, a Bakry-Qian inequality, the Harnack inequality.

math.AP↗

arXiv2013

Immersed Spheres of Finite Total Curvature into Manifolds

We prove that a sequence of possibly branched, weak immersions of the two-sphere $S^2$ into an arbitrary compact riemannian manifold $(M^m,h)$ with uniformly bounded area and uniformly bounded $L^2-$norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a subsequence, to a Lipschitz mapping of $S^2$ and whose image is made of a connected union of finitely many, possibly branched, weak immersions of $S^2$ with finite total curvature. We prove moreover that if the sequence belongs to a class $γ$ of $π_2(M^m)$ the limiting lipschitz mapping of $S^2$ realizes this class as well.

math.DG↗

arXiv2013

Embedded surfaces of arbitrary genus minimizing the Willmore energy under isoperimetric constraint

The isoperimetric ratio of an embedded surface in $R^3$ is defined as the ratio of the area of the surface to power three to the squared enclosed volume. The aim of the present work is to study the minimization of the Willmore energy under fixed isoperimetric ratio when the underlying abstract surface has fixed genus $g\geq 0$. The corresponding problem in the case of spherical surfaces, i.e. $g=0$, was recently solved by Schygulla with different methods.

math.DG↗

arXiv2013

Euclidean spaces as weak tangents of infinitesimally Hilbertian metric spaces with Ricci curvature bounded below

We show that in any infinitesimally Hilbertian $CD^*(K,N)$-space at almost every point there exists a Euclidean weak tangent, i.e. there exists a sequence of dilations of the space that converges to a Euclidean space in the pointed measured Gromov-Hausdorff topology. The proof follows by considering iterated tangents and the splitting theorem for infinitesimally Hilbertian $CD^*(0,N)$-spaces.

math.MG↗

arXiv2013

Existence of generalized totally umbilic 2-spheres in perturbed 3-spheres

It was recently shown by R. Souam and E. Toubiana that the (non constantly curved) Berger spheres do not contain totally umbilic surfaces. Nevertheless in this article we show, by perturbative arguments, that all analytic metrics sufficiently close to the round metric $g_{0}$ on $\mathbb{S}^{3}$ possess \textsl{generalized} totally umbilic 2-spheres, namely critical points of the conformal Willmore functional $\int_Σ|A^\circ|^{2}\,dμ_γ$. The same is true in the smooth setting provided a suitable non-degeneracy condition on the traceless Ricci tensor holds. The proof involves a gluing process of two different finite-dimensional reduction schemes, a sharp asymptotic analysis of the functional on perturbed umbilic spheres of small radius and a quantitative Schur-type Lemma in order to treat the cases when the traceless Ricci tensor of the perturbation is degenerate but not identically zero. For left-invariant metrics on $SU(2)\cong \mathbb{S}^{3}$ our result implies the existence of uncountably many distinct Willmore spheres.

math.DG↗

arXiv2013

Riemannian Ricci curvature lower bounds in metric measure spaces with $σ$-finite measure

Using techniques of optimal transportation and gradient flows in metric spaces, we extend the notion of Riemannian Curvature Dimension condition $RCD(K,\infty)$ introduced (in case the reference measure is finite) by Giuseppe Savare', the first and the second author, to the case the reference measure is $σ$-finite; in this way the theory includes natural examples as the euclidean $n$-dimensional space endowed with the Lebesgue measure, and noncompact manifolds with bounded geometry endowed with the Riemannian volume measure. Another major goal of the paper is to simplify the axiomatization of $RCD(K,\infty)$ (even in case of finite reference measure) replacing the assumption of strict $CD(K,\infty)$ with the classic notion of $CD(K,\infty)$.

math.DG↗

arXiv2013

A new notion of angle between three points in a metric space

We give a new notion of angle in general metric spaces; more precisely, given a triple a points $p,x,q$ in a metric space $(X,d)$, we introduce the notion of angle cone ${\angle_{pxq}}$ as being an interval ${\angle_{pxq}}:=[\angle^-_{pxq},\angle^+_{pxq}]$, where the quantities $\angle^\pm_{pxq}$ are defined in terms of the distance functions from $p$ and $q$ via a duality construction of differentials and gradients holding for locally Lipschitz functions on a general metric space. Our definition in the Euclidean plane gives the standard angle between three points and in a Riemannian manifold coincides with the usual angle between the geodesics, if $x$ is not in the cut locus of $p$ or $q$. We show that in general the angle cone is not single valued (even in case the metric space is a smooth Riemannian manifold, if $x$ is in the cut locus of $p$ or $q$), but if we endow the metric space with a positive Borel measure $m$ obtaining the metric measure space $(X,d,m)$ then under quite general assumptions (which include many fundamental examples as Riemannian manifolds, finite dimensional Alexandrov spaces with curvature bounded from below, Gromov-Hausdorff limits of Riemannian manifolds with Ricci curvature bounded from below, and normed spaces with strictly convex norm), fixed $p,q \in X$, the angle cone at $x$ is single valued for $m$-a.e. $x \in X$. We prove some basic properties of the angle cone (such as the invariance under homotheties of the space) and we analyze in detail the case $(X,d,m)$ is a measured-Gromov-Hausdorff limit of a sequence of Riemannian manifolds with Ricci curvature bounded from below, showing the consistency of our definition with a recent construction of Honda.

math.MG↗

arXiv2012

Existence of isoperimetric regions in non-compact Riemannian manifolds under Ricci or scalar curvature conditions

We prove existence of isoperimetric regions for every volume in non-compact Riemannian $n$-manifolds $(M,g)$, $n\geq 2$, having Ricci curvature $Ric_g\geq (n-1) k_0 g$ and being locally asymptotic to the simply connected space form of constant sectional curvature $k_0$; moreover in case $k_0=0$ we show that the isoperimetric regions are indecomposable. We also discuss some physically and geometrically relevant examples. Finally, under assumptions on the scalar curvature we prove existence of isoperimetric regions of small volume.

math.DG↗

arXiv2012

A PDE approach to nonlinear potential theory in metric measure spaces

We show that the tools recently introduced by the first author in [9] allow to give a PDE description of p-harmonic functions in metric measure setting. Three applications are given: the first is about new results on the sheaf property of harmonic functions, the second is a PDE proof of the fact that the composition of a subminimizer with a convex and non-decreasing function is again a subminimizer, and the third is the fact that the Busemann function associated to a line is harmonic on infinitesimally Hilbertian CD(0,N) spaces.

math.MG↗

arXiv2012

Willmore Spheres in Compact Riemannian Manifolds

The paper is devoted to the variational analysis of the Willmore, and other L^2 curvature functionals, among immersions of 2-dimensional surfaces into a compact riemannian m-manifold (M^m,h) with m>2. The goal of the paper is twofold, on one hand, we give the right setting for doing the calculus of variations (including min max methods) of such functionals for immersions into manifolds and, on the other hand, we prove existence results for possibly branched Willmore spheres under various constraints (prescribed homotopy class, prescribed area) or under curvature assumptions for M^m. To this aim, using the integrability by compensation, we develop first the regularity theory for the critical points of such functionals. We then prove a rigidity theorem concerning the relation between CMC and Willmore spheres. Then we prove that, for every non null 2-homotopy class, there exists a representative given by a Lipschitz map from the 2-sphere into M^m realizing a connected family of conformal smooth (possibly branched) area constrained Willmore spheres (as explained in the introduction, this comes as a natural extension of the minimal immersed spheres in homotopy class constructed by Sacks and Uhlembeck in \cite{SaU}, in situations when they do not exist). Moreover, for every A>0 we minimize the Willmore functional among connected families of weak, possibly branched, immersions of the 2-sphere having prescribed total area equal to A and we prove full regularity for the minimizer. Finally, under a mild curvature condition on (M^m,h), we minimize the sum of the area with the square of the L^2 norm of the second fundamental form, among weak possibly branched immersions of the two sphere and we prove the regularity of the minimizer.

math.AP↗

arXiv2012

The Conformal Willmore Functional: a Perturbative Approach

The conformal Willmore functional (which is conformal invariant in general Riemannian manifold $(M,g)$) is studied with a perturbative method: the Lyapunov-Schmidt reduction. Existence of critical points is shown in ambient manifolds $(\mathbb{R}^3, g_ε)$ -where $g_ε$ is a metric close and asymptotic to the euclidean one. With the same technique a non existence result is proved in general Riemannian manifolds $(M,g)$ of dimension three.

math.DG↗

arXiv2012

Existence of immersed spheres minimizing curvature functionals in non-compact 3-manifolds

We study curvature functionals for immersed 2-spheres in non-compact, three-dimensional Riemannian manifold $(M,h)$ without boundary. First, under the assumption that $(M,h)$ is the euclidean 3-space endowed with a semi-perturbed metric with perturbation small in $C^1$ norm and of compact support, we prove that if there is some point $\bar{x} \in M$ with scalar curvature $R^M(\bar{x})>0$ then there exists a smooth embedding $f:S^2 \hookrightarrow M$ minimizing the Willmore functional $1/4\int |H|^2$, where $H$ is the mean curvature. Second, assuming that $(M,h)$ is of bounded geometry (i.e. bounded sectional curvature and strictly positive injectivity radius) and asymptotically euclidean or hyperbolic we prove that if there is some point $\bar{x} \in M$ with scalar curvature $R^M(\bar{x})>6$ then there exists a smooth immersion $f:S^2 \hookrightarrow M$ minimizing the functional $\int (1/2|A|^2+1)$, where $A$ is the second fundamental form. Finally, adding the bound $K^M \leq 2$ to the last assumptions, we obtain a smooth minimizer $f:S^2 \hookrightarrow M$ for the functional $\int (1/4|H|^2+1)$. The assumptions of the last two theorems are satisfied in a large class of 3-manifolds arising as spacelike timeslices solutions of the Einstein vacuum equation in case of null or negative cosmological constant.

math.DG↗

arXiv2011

Existence of immersed spheres minimizing curvature functionals in compact 3-manifolds

We study curvature functionals for immersed 2-spheres in a compact, three-dimensional Riemannian manifold M. Under the assumption that the sectional curvature of M is strictly positive, we prove the existence of a smoothly immersed sphere minimizing the L^{2} integral of the second fundamental form. Assuming instead that the sectional curvature is less than or equal to 2, and that there exists a point in M with scalar curvature bigger than 6, we obtain a smooth 2-sphere minimizing the integral of 1/4|H|^{2} +1, where H is the mean curvature vector.

math.DG↗