1,720,964 research outputs found
Geometric aspects of p-capacitary potentials
No full textWe provide monotonicity formulas for solutions to the p-Laplace equation defined in the exterior of a convex domain. A number of analytic and geometric consequences are derived, including the classical Minkowski inequality as well as new characterizations of rotationally symmetric solutions and domains. The proofs rely on the conformal splitting technique introduced by the second author in collaboration with V. Agostiniani
Minimising hulls, p-capacity and isoperimetric inequality on complete Riemannian manifolds
Full text linkThe notion of strictly outward minimising hull is investigated for open sets of finite perimeter sitting inside a complete noncompact Riemannian manifold. Under natural geometric assumptions on the ambient manifold, the strictly outward minimising hull Ω⁎ of a set Ω is characterised as a maximal volume solution of the least area problem with obstacle, where the obstacle is the set itself. In the case where Ω has C1,α-boundary, the area of ∂Ω⁎ is recovered as the limit of the p-capacities of Ω, as p→1+. Finally, building on the existence of strictly outward minimising exhaustions, a sharp isoperimetric inequality is deduced on complete noncompact manifolds with nonnegative Ricci curvature, provided 3≤n≤7
On the isoperimetric Riemannian Penrose inequality
Full text linkWe prove that the Riemannian Penrose inequality holds for asymptotically flat 3-manifolds with nonnegative scalar curvature and connected horizon boundary, provided the optimal decay assumptions are met, which result in the (Formula presented.) mass being a well-defined geometric invariant. Our proof builds on a novel interplay between the Hawking mass and a potential-theoretic version of it, recently introduced by Agostiniani, Oronzio, and the third named author. As a consequence, we establish the equality between (Formula presented.) mass and Huisken's isoperimetric mass under the above sharp assumptions. Moreover, we establish a Riemannian Penrose inequality in terms of the isoperimetric mass on any 3-manifold with nonnegative scalar curvature, connected horizon boundary, and which supports a well-posed notion of weak inverse mean curvature flow (IMCF). In particular, such isoperimetric Riemannian Penrose inequality does not require the asymptotic flatness of the manifold. The argument is based on a new asymptotic comparison result involving Huisken's isoperimetric mass and the Hawking mass
The asymptotic behaviour of p-capacitary potentials in asymptotically conical manifolds
No full textWe study the asymptotic behaviour of the p-capacitary potential and of the weak inverse mean curvature flow of a bounded set along the ends of an asymptotically conical Riemannian manifolds with asymptotically nonnegative Ricci curvature
Comparing monotonicity formulas for electrostatic potentials and static metrics
Full text linkIn this note we survey and compare the monotonicity formulas recently discovered by the authors in [1] and [2] in the context of classical potential theory and in the study of static metrics, respectively. In both cases we discuss the most significant implications of the monotonicity formulas in terms of sharp analytic and geometric inequalities. In particular, we derive the classical Willmore inequality for smooth compact hypersurfaces embedded in Euclidean space and the Riemannian Penrose inequality for static Black Holes with connected horizon
Nonlinear Isocapacitary Concepts of Mass in 3-Manifolds with Nonnegative Scalar Curvature
Full text linkWe deal with suitable nonlinear versions of Jauregui's isocapacitary mass in 3-manifolds with nonnegative scalar curvature and compact outermost minimal boundary. These masses, which depend on a parameter 1 1(+). We derive positive mass theorems for these masses under mild conditions at infinity, and we show that these masses do coincide with the ADM mass when the latter is defined. We finally work out a nonlinear potential theoretic proof of the Penrose inequality in the optimal asymptotic regime
Minkowski Inequalities via Nonlinear Potential Theory
Full text linkIn this paper, we prove an extended version of the Minkowski Inequality, holding for any smooth bounded set Ω ⊂ Rn, n≥ 3. Our proof relies on the discovery of effective monotonicity formulas holding along the level set flow of the p-capacitary potentials associated with Ω , for every p sufficiently close to 1. These formulas also testify the existence of a link between the monotonicity formulas derived by Colding and Minicozzi for the level set flow of Green’s functions and the monotonicity formulas employed by Huisken, Ilmanen and several other authors in studying the geometric implications of the Inverse Mean Curvature Flow. In dimension n≥ 8 , our conclusions are stronger than the ones obtained so far through the latter mentioned technique
Correction: A Note on the Critical Laplace Equation and Ricci Curvature (The Journal of Geometric Analysis, (2023), 33, 6, (178), 10.1007/s12220-023-01223-y)
No full textMINKOWSKI INEQUALITY ON COMPLETE RIEMANNIAN MANIFOLDS WITH NONNEGATIVE RICCI CURVATURE
Full text linkWe consider Riemannian manifolds of dimension at least 3, with nonnegative Ricci curvature and Euclidean volume growth. For every open bounded subset with smooth boundary we establish the validity of an optimal Minkowski inequality. We also characterise the equality case, provided the domain is strictly outward minimising and strictly mean convex. Along with the proof, we establish in full generality sharp monotonicity formulas, holding along the level sets of p-capacitary potentials in p-nonparabolic manifolds with nonnegative Ricci curvature
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