Noncompact Riemannian Research Articles

Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products

Abstract Our purpose is to establish nonexistence results concerning complete noncompact mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products, under mild constraints on the warping and soliton functions. Applications to self-shrinkers in the Euclidean space, as well as to mean curvature flow solitons in other important warped product models such as the Schwarzschild and Reissner-Nordström spaces, and Robertson-Walker spacetimes such as the Einstein-de Sitter spacetime, are also given. Furthermore, we study the nonexistence of entire solutions to the mean curvature flow equation. Our approach is based on a suitable conformal change of metric jointly with a maximum principle for complete noncompact Riemannian manifolds with polynomial volume growth due to Alías et al.

Inequalities for eigenvalues of the bi-drifting Laplacian on bounded domains in complete noncompact Riemannian manifolds and related results

Inequalities for eigenvalues of the bi-drifting Laplacian on bounded domains in complete noncompact Riemannian manifolds and related results

Revisiting Gradient Bach Solitons via Maximum Principles

Supposing that the Ricci curvature has an appropriate lower bound and applying suitable maximum principles, we establish triviality results which guarantee that a gradient Bach soliton must be trivial and Bach-flat. Our approach is based on three main cores: convergence to zero at infinity, polynomial volume growth (both related to complete noncompact Riemannian manifolds) and stochastic completeness.

Asymptotic Behavior of Solutions of the Inhomogeneous Schrödinger Equation on Noncompact Riemannian Manifolds

Asymptotic Behavior of Solutions of the Inhomogeneous Schrödinger Equation on Noncompact Riemannian Manifolds

Rigidity and nonexistence of complete hypersurfaces via Liouville type results and other maximum principles, with applications to entire graphs

We investigate complete hypersurfaces with some positive higher order mean curvature in a semi-Riemannian warped product space. Under standard curvature conditions on the ambient space and appropriate constraints on the higher order mean curvatures, we establish rigidity and nonexistence results via Liouville type results and suitable maximum principles related to the divergence of smooth vector fields on a complete noncompact Riemannian manifold. Applications to standard warped product models, like the Schwarzschild, Reissner-Nordström and pseudo-hyperbolic spaces, as well as steady state type spacetimes, are given and a particular study of entire graphs is also presented.

Global sharp gradient estimates for a nonlinear parabolic equation on Riemannian manifolds

Abstract In this paper, we employ the techniques in [C. Cavaterra, S. Dipierro, Z. Gao and E. Valdinoci, Global gradient estimates for a general type of nonlinear parabolic equations, J. Geom. Anal. 32 2022, 2, Paper No. 65] and the approach in [H. T. Dung and N. T. Dung, Sharp gradient estimates for a heat equation in Riemannian manifolds, Proc. Amer. Math. Soc. 147 2019, 12, 5329–5338] to derive sharp gradient estimates for a positive solution to the heat equation u t = Δ ⁢ u + a ⁢ u ⁢ log ⁡ u u_{t}=\Delta u+au\log u in a complete noncompact Riemannian manifold (where a is a real constant). This is an extension of the gradient estimates of Dung and Dung.

Diameter estimates for submanifolds in manifolds with nonnegative curvature

Given a closed connected manifold smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we estimate the intrinsic diameter of the submanifold in terms of its mean curvature field integral. On the other hand, for a compact convex surface with boundary smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we can estimate its intrinsic diameter in terms of its mean curvature field integral and the length of its boundary. These results are supplements of previous work of Topping, Wu-Zheng and Paeng.

Liouville Type Theorems for Nonlinear p-Laplacian Equation on Complete Noncompact Riemannian Manifolds

Liouville Type Theorems for Nonlinear p-Laplacian Equation on Complete Noncompact Riemannian Manifolds

Liouville theorems for ancient solutions to the [formula omitted]-harmonic map heat flows

We investigate the Liouville properties of ancient solutions to the V-harmonic map heat flows from complete noncompact Riemannian manifolds with nonnegative Bakry–Emery Ricci curvature. When the target is a simply connected complete Riemannian manifold with nonpositive sectional curvature, then a Liouville theorem of ancient solutions holds under certain growth condition near infinity. When the target is a complete Riemannian manifold with sectional curvature bounded above by a positive constant, we show that if the image of the ancient solution u is contained in a regular ball near infinity, then u is a constant.

Talenti’s Comparison Theorem for Poisson Equation and Applications on Riemannian Manifold with Nonnegative Ricci Curvature

In this article, we prove Talenti’s comparison theorem for Poisson equation on complete noncompact Riemannian manifold with nonnegative Ricci curvature. Furthermore, we obtain the Faber–Krahn inequality for the first eigenvalue of Dirichlet Laplacian, $$L^1$$ - and $$L^\infty $$ -moment spectrum, especially Saint-Venant theorem for torsional rigidity and a reverse Hölder inequality for eigenfunctions of Dirichlet Laplacian.

Nonexistence for time-fractional wave inequalities on Riemannian manifolds

<p style='text-indent:20px;'>We establish necessary conditions for the existence of global weak solutions to a class of semilinear time-fractional wave inequalities with nonlinearity of derivative type, defined on complete noncompact Riemannian manifolds. A potential function depending of both time and space, is allowed in front of the power nonlinearity. The obtained conditions depend on the parameters of the problem, the initial conditions and the geometry of the manifold. Our results are new even in the Euclidean case.</p>

On the higher order mean curvatures of spacelike hypersurfaces in pp-wave spacetimes

We study some geometric aspects of the higher order mean curvatures (or, more simply, the so-called $ r $-th mean curvatures) of a spacelike hypersurface immersed in a pp-wave spacetime, namely, in a connected Lorentzian manifold admitting a parallel and lightlike vector field. Initially, we develop general Minkowski-type integral formulas for compact (without boundary) spacelike hypersurfaces and we apply them to the study of the uniqueness and nonexistence of compact spacelike hypersurfaces in terms of their $ r $-mean curvatures. Next, we obtain a characterization of $ r $-stability for $ r $-maximal compact spacelike hypersurfaces through of the analysis of the first nonzero eigenvalue of an differential operator naturally attached to the $ r $-th mean curvature. For the noncompact case, by applying new forms of maximum principles on complete noncompact Riemannian manifolds due to Caminha [17] and Alías, Caminha and Nascimento [3], we obtain sufficient geometric conditions involving some $ r $-th mean curvature and the volume growth that allow us to establish some nonexistence results or to guarantee that a complete noncompact spacelike hypersurface is either totally geodesic, or totally umbilical, or maximal, or $ r $-maximal. We also obtain estimates for the index of minimum relative nullity of spacelike hypersurfaces.

Hölder regularity and Liouville properties for nonlinear elliptic inequalities with power-growth gradient terms

This note studies local integral gradient bounds for distributional solutions of a large class of partial differential inequalities with diffusion in divergence form and power-like first-order terms. The applications of these estimates are two-fold. First, we show the (sharp) global Hölder regularity of distributional semi-solutions to this class of diffusive PDEs with first-order terms having supernatural growth and right-hand side in a suitable Morrey class posed on a bounded and regular open set$\Omega$. Second, we provide a new proof of entire Liouville properties for inequalities with superlinear first-order terms without assuming any one-side bound on the solution for the corresponding homogeneous partial differential inequalities. We also discuss some extensions of the previous properties to problems arising in sub-Riemannian geometry and also to partial differential inequalities posed on noncompact complete Riemannian manifolds under appropriate area-growth conditions of the geodesic spheres, providing new results in both these directions. The methods rely on integral arguments and do not exploit maximum and comparison principles.

Nonexistence and umbilicity of spacelike submanifolds with second fundamental form locally timelike

Under the assumption that the second fundamental form is locally timelike, we establish new nonexistence and umbilicity results concerning n-dimensional spacelike submanifolds immersed with parallel mean curvature vector in the (n+p)-dimensional de Sitter space mathbb {S}^{n+p}_q of index q, such that 1le qle p. Our approach is based on a Simon’s type inequality involving the norm of the total umbilicity tensor, obtained by Mariano in [17], jointly with suitable maximum principles due to Alías, Caminha and do Nascimento [6, 7] for complete noncompact Riemannian manifolds and a weak version of Omori–Yau’s maximum principle for stochastically complete Riemanian manifolds proved by Pigola, Rigoli and Setti [20, 21].

Dirac operators on foliations with invariant transverse measures

We extend the groundbreaking results of Gromov and Lawson, [13], to Dirac operators defined along the leaves of foliations of non-compact complete Riemannian manifolds which admit invariant transverse measures. We prove a relative measured index theorem for pairs of such manifolds, foliations and operators, which are identified off compact subsets of the manifolds. We assume that the spectral projections of the leafwise operators for some interval [0,ϵ], ϵ>0, have finite dimensional images when paired with the invariant transverse measures. As a prime example, we show that if the zeroth order operators in the associated Bochner Identities are uniformly positive off compact subsets of the manifolds, then they satisfy the hypotheses of our relative measured index theorem. Using these results, we show that for a large collection of spin foliations, the space of positive scalar curvature metrics on each foliation has infinitely many path connected components.

Elliptic differential inclusions on non-compact Riemannian manifolds

We investigate a large class of elliptic differential inclusions on non-compact complete Riemannian manifolds which involves the Laplace–Beltrami operator and a Hardy-type singular term. Depending on the behavior of the nonlinear term and on the curvature of the Riemannian manifold, we guarantee non-existence and existence/multiplicity of solutions for the studied differential inclusion. The proofs are based on nonsmooth variational analysis as well as isometric actions and fine eigenvalue properties on Riemannian manifolds. The results are also new in the smooth setting.

Volume growth on manifolds with quadratic curvature decay and its applications

In this paper, we establish global volume growth on an n−dimensional complete noncompact Riemannian manifold whose Ricci curvature is bounded from below byRic(x)≥−(n−1)K01+r2(x), where K0 is a positive constant and r(x)=dist(O,x) is the distance function determined by a fixed point O∈M. Based on which, we derive the Lp−almost boundedness of the Riesz transformation for p∈[1,2] on these manifolds.

A Note on Constant Mean Curvature Foliations of Noncompact Riemannian Manifolds

We aimed to study constant mean curvature foliations of noncompact Riemannian manifolds, satisfying some geometric constraints. As a byproduct, we answer a question by M. P. do Carmo (see Introduction) about the leaves of such foliations.

Nonnegative scalar curvature and area decreasing maps on complete foliated manifolds

Abstract Let ( M , g T ⁢ M ) {(M,g^{TM})} be a noncompact complete Riemannian manifold of dimension n, and let F ⊆ T ⁢ M {F\subseteq TM} be an integrable subbundle of TM. Let g F = g T ⁢ M | F {g^{F}=g^{TM}|_{F}} be the restricted metric on F and let k F {k^{F}} be the associated leafwise scalar curvature. Let f : M → S n ⁢ ( 1 ) {f:M\to S^{n}(1)} be a smooth area decreasing map along F, which is locally constant near infinity and of non-zero degree. We show that if k F > rk ⁢ ( F ) ⁢ ( rk ⁢ ( F ) - 1 ) {k^{F}>{\rm rk}(F)({\rm rk}(F)-1)} on the support of d ⁢ f {{\rm d}f} , and either TM or F is spin, then inf ⁡ ( k F ) < 0 {\inf(k^{F})<0} . As a consequence, we prove Gromov’s sharp foliated ⊗ ε {\otimes_{\varepsilon}} -twisting conjecture. Using the same method, we also extend two famous non-existence results due to Gromov and Lawson about Λ 2 {\Lambda^{2}} -enlargeable metrics (and/or manifolds) to the foliated case.

Minimising hulls, p-capacity and isoperimetric inequality on complete Riemannian manifolds

The 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.