arXiv subjects

Marco Rigoli

Publications and source records attributed to Marco Rigoli.

At least 19 recordsLinked to original sources

arXiv2026

On the Geometry of Cotton Gravity

We analyze the geometry of the field equations of Cotton gravity (for a quite general energy-momentum tensor) on a static space-time. In particular, we describe the local structure of the spatial Riemannian factor. This structure, that we call Cotton-$\varphi$-perfect fluid (C-$\varphi$-PF, for short) is a generalization to the regime of Cotton Gravity of the recently introduced notion of $\varphi$-static perfect fluid space-time ($\varphi$-SPFST). After discussing the variational origin of this system, we provide sufficient conditions for a C-$\varphi$-PF to reduce to a $\varphi$-SPFST. We also study the geometry of the level sets of the lapse function $f$ and we provide a rigidity result for C-$\varphi$-PFs under some curvature conditions. The role that Codazzi tensors hold in this theory is highlighted.

gr-qc↗

arXiv2025

On the classification of capillary graphs in Euclidean and non-Euclidean spaces

We prove some rigidity and classification results for graphs with prescribed mean curvature and locally constant Dirichlet and Neumann data, for instance as they appear in capillarity problems. We consider domains in Riemannian manifolds, with emphasis on $\mathbb{R}^2$ and $\mathbb{R}^3$. We classify both the underlying domain and the resulting solution, providing general splitting theorems in this setting.

math.DG↗

arXiv2025

Proper solutions of the $1/H$-flow and the Green kernel of the $p$-Laplacian

We show existence and optimal growth estimate for the weak inverse mean curvature flow issuing from a point, on manifolds with certain curvature and isoperimetric conditions. These theorems imply analogous ones for the flow issuing from relatively compact sets. Some of the results are obtained by proving new decay estimates for the Green kernel of the $p$-Laplacian which fix a gap in the literature. Additionally, we address the convergence of renormalized $p$-capacitary potentials to the inverse mean curvature flow with outer obstacle.

math.AP↗

arXiv2025

On the Geometry of $\varphi$-Static Perfect Fluid Space-Times

In this paper we study the geometry of $\varphi$-static perfect fluid space-times ($\varphi$-SPFST, for short). In the context of Einstein's General Relativity, they arise from a space-time whose matter content is described by a perfect fluid in addition to a nonlinear field expressed by a smooth map $\varphi$ with values in a Riemannian manifold. Considering the Lorentzian manifold $\hat{M}$ in the form of a static warped product, we derive the fundamental equations via reduction of Einstein's Field Equations to the factors of the product. To set the stage for our main results, we discuss the validity of the classical Energy Conditions in the present setting and we introduce the formalism of $\varphi$-curvatures, which is a fundamental tool to merge the geometry of the manifold with that of the smooth map $\varphi$. We then present several mathematical settings in which similar structures arise. After computing two integrability conditions, we apply them to prove a number of rigidity results, both for manifolds with or without boundary. In each of the aforementioned results, the main assumption is given by the vanishing of some $\varphi$-curvature tensors and the conclusion is a local splitting of the metric into a warped product. Inspired by the classical Cosmic No Hair Conjecture of Boucher, Gibbons and Horowitz, we find sharp sufficient conditions on a compact $\varphi$-SPFST with boundary to be isometric to the standard hemisphere. We then describe the geometry of relatively compact domains in $M$ subject to an upper bound on the mean curvature of their boundaries. Finally, we study non-existence results for $\varphi$-SPFSTs, both via the existence of zeroes of the solutions of an appropriate ODE and with the aid of a suitable integral formula generalising in a precise sense the well-known Kazdan-Warner obstruction.

math.DG↗

arXiv2025

Some geometric properties of generalized $\varphi$-vacuum static spaces

In this paper we study the geometry of generalized $\varphi$-vacuum static spaces, proving estimates for the $\varphi$-scalar curvature and for the first eigenvalue of the Jacobi operator, and also rigidity under various geometric assumptions; in particular, we prove a result related to the famous Cosmic no-hair conjecture of Boucher, Gibbons and Horowitz.

math.DG↗

arXiv2024

Some splitting and rigidity results for sub-static spaces

In this paper we study the rigidity problem for sub-static systems with possibly non-empty boundary. First, we get local and global splitting theorems by assuming the existence of suitable compact minimal hypersurfaces, complementing recent results in the literature. Next, we prove some boundary integral inequalities that extend works by Chr\'usciel and Boucher-Gibbons-Horowitz to non-vacuum spaces. Even in the vacuum static case, the inequalities improve on known ones. Lastly, we consider the system arising from static solutions to the Einstein field equations coupled with a $\sigma$-model. The Liouville theorem we obtain allows for positively curved target manifolds, generalizing a result by Reiris.

math.DG↗

arXiv2024

A sharp Eells-Sampson type theorem under positive sectional curvature upper bounds

We prove an extension of Eells and Sampson's rigidity theorem for harmonic maps from a closed manifold of non-negative Ricci curvature to a manifold of non-positive sectional curvature. We give an application of our result in the setting of harmonic-Einstein (or Ricci-harmonic) metrics and as a consequence we recover a classical rigidity result of Hamilton for the problem of prescribed positive definite Ricci curvature.

math.DG↗

arXiv2024

Maximum principles for weakly $1$-coercive operators with applications to capillary and prescribed mean curvature graphs

In this paper we establish maximum principles for weakly 1-coercive operators $L$ on complete, non-compact Riemannian manifolds $M$. In particular, we search for conditions under which one can guarantee that solutions $u$ of differential equations of the form $L(u)\geq f(u)$ satisfy $f(u)\leq 0$ on $M$. The case of weakly $p$-coercive operators with $p>1$, including the $p$-Laplacian and in particular the Laplace-Beltrami operator for $p=2$, has been considered in a recent paper of ours. As a consequence of the main results we infer comparison principles for that kind of operators. Furthermore we apply them to geometric situations dealing with the mean curvature operator, which is a typical weakly 1-coercive operator. We first consider the case of $\mathcal C^1$ operators $L$ acting on functions $u$ of class $\mathcal C^2$ and, in the last section of the paper, we show how our results can be extended to the case of less regular operators $L$ acting on functions $u$ which are just continuous and locally $W^{1,1}$ regular.

math.AP↗

arXiv2023

On minimal graphs of sublinear growth over manifolds with non-negative Ricci curvature

We prove that entire solutions of the minimal hypersurface equation \[ \mathrm{div}\left(\frac{Du}{\sqrt{1+|Du|^2}}\right) = 0 \] on a complete manifold with $\mathrm{Ric} \ge 0$, whose negative part grows like $\mathcal{O}(r/\log r)$ ($r$ the distance from a fixed origin), are constant. This extends the Bernstein Theorem for entire positive minimal graphs established in recent years. The proof depends on a new technique to get gradient bounds by means of integral estimates, which does not require any further geometric assumption on $M$.

math.DG↗

arXiv2023

On the $1/H$-flow by $p$-Laplace approximation: new estimates via fake distances under Ricci lower bounds

In this paper we show the existence of weak solutions $w : M \rightarrow \mathbb{R}$ of the inverse mean curvature flow starting from a relatively compact set (possibly, a point) on a large class of manifolds satisfying Ricci lower bounds. Under natural assumptions, we obtain sharp estimates for the growth of $w$ and for the mean curvature of its level sets, that are well behaved with respect to Gromov-Hausdorff convergence. The construction follows R. Moser's approximation procedure via the $p$-Laplace equation, and relies on new gradient and decay estimates for $p$-harmonic capacity potentials, notably for the kernel $\mathcal{G}_p$ of $Δ_p$. These bounds, stable as $p \rightarrow 1$, are achieved by studying fake distances associated to capacity potentials and Green kernels. We conclude by investigating some basic isoperimetric properties of the level sets of $w$.

math.DG↗

arXiv2023

Growth of subsolutions of $Δ_p u = V|u|^{p-2}u$ and of a general class of quasilinear equations

In this paper we prove some integral estimates on the minimal growth of the positive part $u_+$ of subsolutions of quasilinear equations \[ \mathrm{div} A(x,u,\nabla u) = V|u|^{p-2}u \] on complete Riemannian manifolds $M$, in the non-trivial case $u_+\not\equiv 0$. Here $A$ satisfies the structural assumption $|A(x,u,\nabla u)|^{p/(p-1)} \leq k \langle A(x,u,\nabla u),\nabla u\rangle$ for some constant $k>0$ and for $p>1$ the same exponent appearing on the RHS of the equation, and $V$ is a continuous positive function, possibly decaying at a controlled rate at infinity. We underline that the equation may be degenerate and that our arguments do not require any geometric assumption on $M$ beyond completeness of the metric. From these results we also deduce a Liouville-type theorem for sufficiently slowly growing solutions.

math.AP↗

arXiv2022

Einstein-type structures, Besse's conjecture and a uniqueness result for a $φ$-CPE metric in its conformal class

In this paper, we study an extension of the CPE conjecture to manifolds $M$ which support a structure relating curvature to the geometry of a smooth map $φ: M \to N$. The resulting system, denoted by $(φ-\mathrm{CPE})$, is natural from the variational viewpoint and describes stationary points for the integrated $φ$-scalar curvature functional restricted to metrics with unit volume and constant $φ$-scalar curvature. We prove both a rigidity statement for solutions to $(φ-\mathrm{CPE})$ in a conformal class, and a gap theorem characterizing the round sphere among manifolds supporting $(φ-\mathrm{CPE})$ with $φ$ a harmonic map.

math.DG↗

arXiv2022

Tachibana-type theorems on complete manifolds

We prove that a compact Riemannian manifold of dimension $m \geq 3$ with harmonic curvature and $\lfloor\frac{m-1}{2}\rfloor$-positive curvature operator has constant sectional curvature, extending the classical Tachibana theorem for manifolds with positive curvature operator. The condition of $\lfloor\frac{m-1}{2}\rfloor$-positivity originates from recent work of Petersen and Wink, who proved a similar Tachibana-type theorem under the stronger condition that the manifold be Einstein. We show that the same rigidity property holds for complete manifolds assuming either parabolicity, an integral bound on the Weyl tensor or a stronger pointwise positive lower bound on the average of the first $\lfloor\frac{m-1}{2}\rfloor$ eigenvalues of the curvature operator. For 3-manifolds, we show that positivity of the curvature operator can be relaxed to positivity of the Ricci tensor.

math.DG↗

arXiv2022

Bernstein and half-space properties for minimal graphs under Ricci lower bounds

In this paper, we prove a new gradient estimate for minimal graphs defined on domains of a complete manifold with Ricci curvature bounded from below. In particular, we show that positive, entire minimal graphs on manifolds with non-negative Ricci curvature are constant, and that complete, parabolic manifolds with Ricci curvature bounded from below have the half-space property. We avoid the need of sectional curvature bounds on $M$ by exploiting a form of the Ahlfors-Khas'minskii duality in nonlinear potential theory.

math.DG↗

arXiv2021

Non-negative Ricci curvature and Minimal graphs with linear growth

We study minimal graphs with linear growth on complete manifolds $M^m$ with $\mathrm{Ric} \ge 0$. Under the further assumption that the $(m-2)$-th Ricci curvature in radial direction is bounded below by $C r(x)^{-2}$, we prove that any such graph, if non-constant, forces tangent cones at infinity of $M$ to split off a line. Note that $M$ is not required to have Euclidean volume growth. We also show that $M$ may not split off any line. Our result parallels that obtained by Cheeger, Colding and Minicozzi for harmonic functions. The core of the paper is a new refinement of Korevaar's gradient estimate for minimal graphs, together with heat equation techniques.

math.DG↗

arXiv2021

Remarks on mean curvature flow solitons in warped products

We study some properties of mean curvature flow solitons in general Riemannian manifolds and in warped products, with emphasis on constant curvature and Schwarzschild type spaces. We focus on splitting and rigidity results under various geometric conditions, ranging from the stability of the soliton to the fact that the image of its Gauss map be contained in suitable regions of the sphere. We also investigate the case of entire graphs.

math.DG↗

arXiv2021

A note on spacelike hypersurfaces and timelike conformal vectors

Any compact spacelike hypersurface immersed in a doubly warped product spacetime $I{}_{h} \times_ρ \mathbb{P}$ with nondecreasing warping factor $ρ$ must be a spacelike slice, provided that the mean curvature satisfies $H\geqρ'/hρ$ everywhere on the hypersurface. The conclusion also holds, under suitable assumptions on the immersion, when the hypersurface is complete and noncompact. A similar rigidity property is shown for compact hypersurfaces in spacetimes carrying a conformal, strictly expanding, timelike vector field.

math.DG↗

arXiv2021

Constant mean curvature spacelike hypersurfaces in spatially open GRW spacetimes

In this paper we provide under certain geometric and physical assumptions new uniqueness and non-existence results for complete spacelike hypersurfaces of constant mean curvature in spatially open Generalized Robertson-Walker spacetimes. Some of our results are then applied to relevant spacetimes as the steady state spacetime, Einstein-de Sitter spacetime and certain radiation models.

math.DG↗