Any compact spacelike hypersurface immersed in a doubly warped product spacetime I h × ρ P with n... more Any compact spacelike hypersurface immersed in a doubly warped product spacetime I h × ρ P with nondecreasing warping factor ρ must be a spacelike slice, provided that the mean curvature satisfies H ≥ ρ ′ /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. 2 The geometric setting Let ψ : (M m , g) → (M m+1 , g) be an isometric immersion between (connected) semi-Riemannian manifolds and denote with ∇ and ∇, respectively, the Levi-Civita connections for g and g. For any given point p ∈ M , there exist sufficiently small neighbourhoods U ⊆ M and U ⊆ M , respectively, of p and ψ(p), such that ψ(U) ⊆ U , ψ| U : U → U is an embedding and U supports a nowhere vanishing vector field Z ∈ X(U) satisfying g(Z ψ(q) , (dψ) q V) = 0 for every q ∈ U , V ∈ T q M. We say that
In this paper we characterize compact and complete hypersurfaces with some constant higher order ... more In this paper we characterize compact and complete hypersurfaces with some constant higher order mean curvature into warped product spaces. Our approach is based on the use of a new trace operator version of the Omori-Yau maximum principle which seems to be interesting in its own.
In this paper we prove some integral estimates on the minimal growth of the positive part u + of ... more In this paper we prove some integral estimates on the minimal growth of the positive part u + of subsolutions of quasilinear equations divA(x, u, ∇u) = V |u| p−2 u on complete Riemannian manifolds M , in the non-trivial case u + ≡ 0. Here A satisfies the structural assumption |A(x, u, ∇u)| p/(p−1) ≤ k A(x, u, ∇u), ∇u 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.
In this paper we establish a natural framework for the stability of mean curvature flow solitons ... more In this paper we establish a natural framework for the stability of mean curvature flow solitons in warped product spaces. These solitons are regarded as stationary immersions for a weighted volume functional. Under this point of view, we are able to find geometric conditions for finiteness of the index and some characterizations of stable solitons. We also prove some non-existence results for solitons as applications of a comparison principle which suits well the structure of the diffusion elliptic operator associated to the weighted measures we are considering.
In this paper we show the existence of weak solutions ∶ → ℝ to the inverse mean curvature flow st... more In this paper we show the existence of weak solutions ∶ → ℝ to 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 and for the mean curvature of its level sets, which are well behaved with respect to Gromov-Hausdorff convergence. The construction follows R. Moser's approximation procedure via the-Laplace equation, and relies on new gradient and decay estimates for-harmonic capacity potentials, notably for the kernel of Δ. These bounds, stable as → 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 .
Nonlinear Analysis-theory Methods & Applications, Mar 1, 2021
The aim of the paper is to study the geometry of a Riemannian manifold M , with a special structu... more The aim of the paper is to study the geometry of a Riemannian manifold M , with a special structure depending on 3 real parameters, a smooth map φ into a target Riemannian manifold N , and a smooth function f on M itself. We will occasionally let some of the parameters be smooth functions. For a special value of one of them, the structure is obtained by a conformal deformation of a harmonic-Einstein manifold. The setting generalizes various previously studied situations; for instance, Ricci solitons, Ricci harmonic solitons, generalized quasi-Einstein manifolds and so on. One main ingredient of our analysis is the study of certain modified curvature tensors on M , related to the map φ, and to develop a series of results for harmonic-Einstein manifolds that parallel those obtained for Einstein manifolds both some time ago and in the very recent literature. We then turn to locally characterize, via a couple of integrability conditions and mild assumptions on f , the manifold M as a warped product with harmonic-Einstein fibers extending in a very non trivial way a recent result for Ricci solitons. We then consider rigidity and non existence, both in the compact and non-compact cases. This is done via integral formulas and, in the non-compact case, via analytical tools previously introduced by the authors.
Communications in Contemporary Mathematics, Dec 1, 2018
We study codimension two trapped submanifolds contained into the future component of the light co... more We study codimension two trapped submanifolds contained into the future component of the light cone of de Sitter spacetime. For codimension two compact spacelike submanifolds in the light cone we show that they are conformally diffeomorphic to the round sphere. This fact enables us to deduce that the problem of characterizing compact marginally trapped submanifolds into the light cone is equivalent to solving the Yamabe problem on the round sphere, allowing us to obtain our main classification result for such submanifolds.
We study the semilinear equation 2u+a(x) u=b(x) u _ (_>1) on a complete Riemannian manifold. We d... more We study the semilinear equation 2u+a(x) u=b(x) u _ (_>1) on a complete Riemannian manifold. We determine conditions on the coefficients that guarantee existence and nonexistence of positive solutions. A very general uniqueness result is also established. Our main results are valid without explicit curvature assumptions, and appear to be new even in R m .
Journal of Mathematical Analysis and Applications, Mar 1, 2020
Abstract The aim of this paper is to continue the geometric-analytic study of φ-curvatures initia... more Abstract The aim of this paper is to continue the geometric-analytic study of φ-curvatures initiated in [2] . These curvatures arise naturally in many geometric contexts, notably in Ricci-harmonic solitons theory. In the present paper we prove two rigidity results related to harmonic-Einstein manifolds, a generalization of the notion of Einstein manifolds to the present situation. We observe that, when we restrict our theorems to the classical case of Einstein manifolds, we obtain some new results even in this setting.
Discrete and Continuous Dynamical Systems - Series S, 2023
We provide full details of the proof of Theorem 3.4 (Theorem C in the Introduction) in Colombo, M... more We provide full details of the proof of Theorem 3.4 (Theorem C in the Introduction) in Colombo, Mari and Rigoli, Remarks on mean curvature flow solitons in warped products. Discrete Contin. Dyn. Syst. Ser. S 13 (2020), no. 7, 1957-1991.
ABSTRACT We introduce a class of “differential operators” on graphs and we prove an energy estima... more ABSTRACT We introduce a class of “differential operators” on graphs and we prove an energy estimate and a Liouville type theorem depending on some structural properties of the operators considered.
As anticipated in the final part of the introduction to Chap. 3, the aim of this chapter is to pr... more As anticipated in the final part of the introduction to Chap. 3, the aim of this chapter is to prove the validity of the weak maximum principle for a large class of operators under the sole assumption of a controlled volume growth of geodesic balls related to the structure of the operator “at infinity”. In doing so, we provide a second a priori estimate for solutions of certain differential inequalities, that, as an application given at the end of the chapter, enables us to generalize a Liouville-type result due, for the case of the Laplacian, to Dancer and Du [97] (see also the previous work by Aronson and Weinberger [32]). We also localize the principle to the family of the open sets with nonempty boundary of the manifold. This new formulation reminds of the (weak form of the) maximum principle as it appears, for instance, in the classical books by Protter and Weinberger [233], Gilbarg and Trudinger [125], or in the more recent work by Pucci and Serrin [235]. We underline its importance by giving an analytical application to the uniqueness problem for the positive solutions of certain Lichnerowicz-type equations.
This chapter is basically devoted to miscellany applications of the results presented in Chaps. 3... more This chapter is basically devoted to miscellany applications of the results presented in Chaps. 3 and 4. We show how, with the aid of various forms of the maximum principle, we can improve on some classical results. In fact we begin with some introductory considerations to motivate a nonimmersibility result for a manifold M into cones of the Euclidean space due to Omori [210] and of which we provide an improved version in Theorem 5.1. We recall that, in the cited work of Omori, we have the first appearance of what is now known in the literature as the Omori-Yau maximum principle. We then continue our investigation by establishing a quantitative estimate, according to the results presented in [181], for the width of the cone of \(\mathbb{R}^{n}\) containing the image of M under a smooth map, see Theorem 5.2. Later, we elaborate on some old result of Jorge and Koutroufiotis [154], (see Theorem 5.6) and we provide a “quantitative” version for immersions into a cone with the aid of the WMP for the Hessian (see Theorem 5.7). With the help of this result and of the theory of flat bilinear forms we are able to consider also the case where M is a Kahler manifold in Corollary 5.7.
The aim of this chapter is to introduce the Omori-Yau and the weak maximum principles. We begin w... more The aim of this chapter is to introduce the Omori-Yau and the weak maximum principles. We begin with some analytical motivations of a general nature and we then proceed to introduce the various concepts, results and related discussions. In this process we follow a perspective quite different from the original approach of Omori and Yau. Indeed, we introduce a function theoretic formulation of the principle that does not tie it to curvature assumptions as in the pioneering works of Omori [210] and Yau [279] (see for instance the statement of Theorem 2.4 below). This formulation reaches a great advantage in applications as it will become crystal clear from the geometric and analytic results contained in the subsequent chapters. We then relax the original statement of the principle to obtain what we call the weak maximum principle.
Annali di Matematica Pura ed Applicata, Jul 16, 2018
We prove higher order mean curvature estimates for properly immersed φ h-bounded hypersurfaces M ... more We prove higher order mean curvature estimates for properly immersed φ h-bounded hypersurfaces M m → (N × L) m+1. The class of φ h-bounded submanifolds, introduced by Bessa et al. (Annali di Matematica Pura ed Applicata 194(1):109-130, 2015), contains properly the class of cylindrically bounded submanifolds. The estimates we prove here depend only on the geometry of N and L and extend previous results of Alias et al.
This chapter is devoted to a quick review of some results in Riemannian geometry using the moving... more This chapter is devoted to a quick review of some results in Riemannian geometry using the moving frame formalism. While we assume basic knowledge of the general subject as presented, for instance, in the standard references [51, 121, 156, 170, 171, 219, 272], several computations will be carried out in full detail in order to acquaint the reader with notation.
In this paper we provide under certain geometric and physical assumptions new uniqueness and non-... more 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.
Any compact spacelike hypersurface immersed in a doubly warped product spacetime I h × ρ P with n... more Any compact spacelike hypersurface immersed in a doubly warped product spacetime I h × ρ P with nondecreasing warping factor ρ must be a spacelike slice, provided that the mean curvature satisfies H ≥ ρ ′ /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. 2 The geometric setting Let ψ : (M m , g) → (M m+1 , g) be an isometric immersion between (connected) semi-Riemannian manifolds and denote with ∇ and ∇, respectively, the Levi-Civita connections for g and g. For any given point p ∈ M , there exist sufficiently small neighbourhoods U ⊆ M and U ⊆ M , respectively, of p and ψ(p), such that ψ(U) ⊆ U , ψ| U : U → U is an embedding and U supports a nowhere vanishing vector field Z ∈ X(U) satisfying g(Z ψ(q) , (dψ) q V) = 0 for every q ∈ U , V ∈ T q M. We say that
In this paper we characterize compact and complete hypersurfaces with some constant higher order ... more In this paper we characterize compact and complete hypersurfaces with some constant higher order mean curvature into warped product spaces. Our approach is based on the use of a new trace operator version of the Omori-Yau maximum principle which seems to be interesting in its own.
In this paper we prove some integral estimates on the minimal growth of the positive part u + of ... more In this paper we prove some integral estimates on the minimal growth of the positive part u + of subsolutions of quasilinear equations divA(x, u, ∇u) = V |u| p−2 u on complete Riemannian manifolds M , in the non-trivial case u + ≡ 0. Here A satisfies the structural assumption |A(x, u, ∇u)| p/(p−1) ≤ k A(x, u, ∇u), ∇u 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.
In this paper we establish a natural framework for the stability of mean curvature flow solitons ... more In this paper we establish a natural framework for the stability of mean curvature flow solitons in warped product spaces. These solitons are regarded as stationary immersions for a weighted volume functional. Under this point of view, we are able to find geometric conditions for finiteness of the index and some characterizations of stable solitons. We also prove some non-existence results for solitons as applications of a comparison principle which suits well the structure of the diffusion elliptic operator associated to the weighted measures we are considering.
In this paper we show the existence of weak solutions ∶ → ℝ to the inverse mean curvature flow st... more In this paper we show the existence of weak solutions ∶ → ℝ to 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 and for the mean curvature of its level sets, which are well behaved with respect to Gromov-Hausdorff convergence. The construction follows R. Moser's approximation procedure via the-Laplace equation, and relies on new gradient and decay estimates for-harmonic capacity potentials, notably for the kernel of Δ. These bounds, stable as → 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 .
Nonlinear Analysis-theory Methods & Applications, Mar 1, 2021
The aim of the paper is to study the geometry of a Riemannian manifold M , with a special structu... more The aim of the paper is to study the geometry of a Riemannian manifold M , with a special structure depending on 3 real parameters, a smooth map φ into a target Riemannian manifold N , and a smooth function f on M itself. We will occasionally let some of the parameters be smooth functions. For a special value of one of them, the structure is obtained by a conformal deformation of a harmonic-Einstein manifold. The setting generalizes various previously studied situations; for instance, Ricci solitons, Ricci harmonic solitons, generalized quasi-Einstein manifolds and so on. One main ingredient of our analysis is the study of certain modified curvature tensors on M , related to the map φ, and to develop a series of results for harmonic-Einstein manifolds that parallel those obtained for Einstein manifolds both some time ago and in the very recent literature. We then turn to locally characterize, via a couple of integrability conditions and mild assumptions on f , the manifold M as a warped product with harmonic-Einstein fibers extending in a very non trivial way a recent result for Ricci solitons. We then consider rigidity and non existence, both in the compact and non-compact cases. This is done via integral formulas and, in the non-compact case, via analytical tools previously introduced by the authors.
Communications in Contemporary Mathematics, Dec 1, 2018
We study codimension two trapped submanifolds contained into the future component of the light co... more We study codimension two trapped submanifolds contained into the future component of the light cone of de Sitter spacetime. For codimension two compact spacelike submanifolds in the light cone we show that they are conformally diffeomorphic to the round sphere. This fact enables us to deduce that the problem of characterizing compact marginally trapped submanifolds into the light cone is equivalent to solving the Yamabe problem on the round sphere, allowing us to obtain our main classification result for such submanifolds.
We study the semilinear equation 2u+a(x) u=b(x) u _ (_>1) on a complete Riemannian manifold. We d... more We study the semilinear equation 2u+a(x) u=b(x) u _ (_>1) on a complete Riemannian manifold. We determine conditions on the coefficients that guarantee existence and nonexistence of positive solutions. A very general uniqueness result is also established. Our main results are valid without explicit curvature assumptions, and appear to be new even in R m .
Journal of Mathematical Analysis and Applications, Mar 1, 2020
Abstract The aim of this paper is to continue the geometric-analytic study of φ-curvatures initia... more Abstract The aim of this paper is to continue the geometric-analytic study of φ-curvatures initiated in [2] . These curvatures arise naturally in many geometric contexts, notably in Ricci-harmonic solitons theory. In the present paper we prove two rigidity results related to harmonic-Einstein manifolds, a generalization of the notion of Einstein manifolds to the present situation. We observe that, when we restrict our theorems to the classical case of Einstein manifolds, we obtain some new results even in this setting.
Discrete and Continuous Dynamical Systems - Series S, 2023
We provide full details of the proof of Theorem 3.4 (Theorem C in the Introduction) in Colombo, M... more We provide full details of the proof of Theorem 3.4 (Theorem C in the Introduction) in Colombo, Mari and Rigoli, Remarks on mean curvature flow solitons in warped products. Discrete Contin. Dyn. Syst. Ser. S 13 (2020), no. 7, 1957-1991.
ABSTRACT We introduce a class of “differential operators” on graphs and we prove an energy estima... more ABSTRACT We introduce a class of “differential operators” on graphs and we prove an energy estimate and a Liouville type theorem depending on some structural properties of the operators considered.
As anticipated in the final part of the introduction to Chap. 3, the aim of this chapter is to pr... more As anticipated in the final part of the introduction to Chap. 3, the aim of this chapter is to prove the validity of the weak maximum principle for a large class of operators under the sole assumption of a controlled volume growth of geodesic balls related to the structure of the operator “at infinity”. In doing so, we provide a second a priori estimate for solutions of certain differential inequalities, that, as an application given at the end of the chapter, enables us to generalize a Liouville-type result due, for the case of the Laplacian, to Dancer and Du [97] (see also the previous work by Aronson and Weinberger [32]). We also localize the principle to the family of the open sets with nonempty boundary of the manifold. This new formulation reminds of the (weak form of the) maximum principle as it appears, for instance, in the classical books by Protter and Weinberger [233], Gilbarg and Trudinger [125], or in the more recent work by Pucci and Serrin [235]. We underline its importance by giving an analytical application to the uniqueness problem for the positive solutions of certain Lichnerowicz-type equations.
This chapter is basically devoted to miscellany applications of the results presented in Chaps. 3... more This chapter is basically devoted to miscellany applications of the results presented in Chaps. 3 and 4. We show how, with the aid of various forms of the maximum principle, we can improve on some classical results. In fact we begin with some introductory considerations to motivate a nonimmersibility result for a manifold M into cones of the Euclidean space due to Omori [210] and of which we provide an improved version in Theorem 5.1. We recall that, in the cited work of Omori, we have the first appearance of what is now known in the literature as the Omori-Yau maximum principle. We then continue our investigation by establishing a quantitative estimate, according to the results presented in [181], for the width of the cone of \(\mathbb{R}^{n}\) containing the image of M under a smooth map, see Theorem 5.2. Later, we elaborate on some old result of Jorge and Koutroufiotis [154], (see Theorem 5.6) and we provide a “quantitative” version for immersions into a cone with the aid of the WMP for the Hessian (see Theorem 5.7). With the help of this result and of the theory of flat bilinear forms we are able to consider also the case where M is a Kahler manifold in Corollary 5.7.
The aim of this chapter is to introduce the Omori-Yau and the weak maximum principles. We begin w... more The aim of this chapter is to introduce the Omori-Yau and the weak maximum principles. We begin with some analytical motivations of a general nature and we then proceed to introduce the various concepts, results and related discussions. In this process we follow a perspective quite different from the original approach of Omori and Yau. Indeed, we introduce a function theoretic formulation of the principle that does not tie it to curvature assumptions as in the pioneering works of Omori [210] and Yau [279] (see for instance the statement of Theorem 2.4 below). This formulation reaches a great advantage in applications as it will become crystal clear from the geometric and analytic results contained in the subsequent chapters. We then relax the original statement of the principle to obtain what we call the weak maximum principle.
Annali di Matematica Pura ed Applicata, Jul 16, 2018
We prove higher order mean curvature estimates for properly immersed φ h-bounded hypersurfaces M ... more We prove higher order mean curvature estimates for properly immersed φ h-bounded hypersurfaces M m → (N × L) m+1. The class of φ h-bounded submanifolds, introduced by Bessa et al. (Annali di Matematica Pura ed Applicata 194(1):109-130, 2015), contains properly the class of cylindrically bounded submanifolds. The estimates we prove here depend only on the geometry of N and L and extend previous results of Alias et al.
This chapter is devoted to a quick review of some results in Riemannian geometry using the moving... more This chapter is devoted to a quick review of some results in Riemannian geometry using the moving frame formalism. While we assume basic knowledge of the general subject as presented, for instance, in the standard references [51, 121, 156, 170, 171, 219, 272], several computations will be carried out in full detail in order to acquaint the reader with notation.
In this paper we provide under certain geometric and physical assumptions new uniqueness and non-... more 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.
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Papers by Marco Rigoli