arXiv subjects

Giulio Colombo

Publications and source records attributed to Giulio Colombo.

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↗

arXiv2026

On $q$-convex hypersurfaces in Riemannian manifolds

We prove that any closed, convex hypersurface in an $(n+1)$-dimensional Riemannian manifold with $\lceil \frac{n}{2} \rceil$-positive curvature operator is a rational homology sphere with finite fundamental group. The same conclusion holds for any $\lceil \frac{n}{2} \rceil$-convex hypersurface, provided that the mean curvature satisfies a sharp pinching condition. Both results follow from more general vanishing and estimation theorems for the Betti numbers of closed $q$-convex immersed hypersurfaces in $(n+1)$-dimensional Riemannian manifolds, under a lower bound on the average of the smallest $(n-p)$ eigenvalues of the curvature operator.

math.DG↗

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

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↗

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

Growth of subsolutions of $\Delta_p u = V|u|^{p-2}u$ 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

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

Einstein-type structures, Besse's conjecture and a uniqueness result for a $\varphi$-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 $\varphi : M \to N$. The resulting system, denoted by $(\varphi-\mathrm{CPE})$, is natural from the variational viewpoint and describes stationary points for the integrated $\varphi$-scalar curvature functional restricted to metrics with unit volume and constant $\varphi$-scalar curvature. We prove both a rigidity statement for solutions to $(\varphi-\mathrm{CPE})$ in a conformal class, and a gap theorem characterizing the round sphere among manifolds supporting $(\varphi-\mathrm{CPE})$ with $\varphi$ a harmonic map.

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

A note on spacelike hypersurfaces and timelike conformal vectors

Any compact spacelike hypersurface immersed in a doubly warped product spacetime $I{}_{h} \times_{\rho} \mathbb{P}$ with nondecreasing warping factor $\rho$ must be a spacelike slice, provided that the mean curvature satisfies $H\geq\rho'/h\rho$ 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↗

arXiv2020

A splitting theorem for capillary graphs under Ricci lower bounds

In this paper, we study capillary graphs defined on a domain $\Omega$ of a complete Riemannian manifold $M$, where a graph is said to be capillary if it has constant mean curvature and locally constant Dirichlet and Neumann conditions on $\partial \Omega$. Our main result is a splitting theorem both for $\Omega$ and for the graph function on a class of manifolds with nonnegative Ricci curvature. As a corollary, we classify capillary graphs over domains that are globally Lipschitz epigraphs or slabs in a product space $M = N \times \mathbb{R}$, where $N$ has slow volume growth and non-negative Ricci curvature, including the case $M = \mathbb{R}^2,\mathbb{R}^3$. A technical core of the paper is a new gradient estimate for positive CMC graphs on manifolds with Ricci lower bounds.

math.DG↗

arXiv2020

Recent rigidity results for graphs with prescribed mean curvature

This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs $u : M \rightarrow \mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical solitons for the mean curvature flow, in warped product ambient spaces. A detailed analysis of the mean curvature operator is given, focusing on maximum principles at infinity, Liouville properties, gradient estimates. Among the geometric applications, we mention the Bernstein theorem for positive entire minimal graphs on manifolds with non-negative Ricci curvature, and a splitting theorem for capillary graphs over an unbounded domain $\Omega \subset M$, namely, for CMC graphs satisfying an overdetermined boundary condition.

math.DG↗

arXiv2019

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↗

arXiv2019

Stable maximal hypersurfaces in Lorentzian spacetimes

We study the geometry of stable maximal hypersurfaces in a variety of spacetimes satisfying various physically relevant curvature assumptions, for instance the Timelike Convergence Condition (TCC). We characterize stability when the target space has constant sectional curvature as well as give sufficient conditions on the geometry of the ambient spacetime (e.g., the validity of TCC) to ensure stability. Some rigidity results and height estimates are also proven in GRW spacetimes. In the last part of the paper we consider $k$-stability of spacelike hypersurfaces, a concept related to mean curvatures of higher orders.

math.DG↗

arXiv2019

Spacelike hypersurfaces in standard static spacetimes

In this work we study spacelike hypersurfaces immersed in spatially open standard static spacetimes with complete spacelike slices. Under appropriate lower bounds on the Ricci curvature of the spacetime in directions tangent to the slices, we prove that every complete CMC hypersurface having either bounded hyperbolic angle or bounded height is maximal. Our conclusions follow from general mean curvature estimates for spacelike hypersurfaces. In case where the spacetime is a Lorentzian product with spatial factor of nonnegative Ricci curvature and sectional curvatures bounded below, we also show that a complete maximal hypersurface not intersecting a spacelike slice is itself a slice. This result is obtained from a gradient estimate for parametric maximal hypersurfaces.

math.DG↗