Complete Noncompact Riemannian Manifold Research Articles

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.

Lord Rayleigh’s Conjecture for Vibrating Clamped Plates in Positively Curved Spaces

We affirmatively solve the analogue of Lord Rayleigh’s conjecture on Riemannian manifolds with positive Ricci curvature for any clamped plates in 2 and 3 dimensions, and for sufficiently large clamped plates in dimensions beyond 3. These results complement those from the flat (Ashbaugh and Benguria in Duke Math J 78(1):1–17, 1995; Nadirashvili in Arch Ration Mech Anal 129(1):1–10, 1995) and negatively curved (Kristály in Adv Math 367:107113, 2020) cases that are valid only in 2 and 3 dimensions, and at the same time also provide the first positive answer to Lord Rayleigh’s conjecture in higher dimensions. The proofs rely on an Ashbaugh–Benguria–Nadirashvili–Talenti nodal-decomposition argument, on the Lévy–Gromov isoperimetric inequality, on fine properties of Gaussian hypergeometric functions and on sharp spectral gap estimates of fundamental tones for both small and large clamped spherical caps. Our results show that positive curvature enhances genuine differences between low- and high-dimensional settings, a tacitly accepted paradigm in the theory of vibrating clamped plates. In the limit case—when the Ricci curvature is non-negative we establish a Lord Rayleigh-type isoperimetric inequality that involves the asymptotic volume ratio of the non-compact complete Riemannian manifold; moreover, the inequality is strongly rigid in 2 and 3 dimensions, i.e., if equality holds for a given clamped plate then the manifold is isometric to the Euclidean space.

Biharmonic maps and biharmonic submanifolds with small curvature integral

In this article, we study biharmonic maps and biharmonic submanifolds with small curvature integral. Let φ:(Mn,g)→(Nm,h) be a biharmonic map from a complete noncompact Riemannian manifold (Mn,g) into a Riemannian manifold (Nm,h) satisfying that the L2-norm of the tension field of the map is finite. If the domain manifold of the map satisfies a Sobolev inequality and the Ln2-norm of the sectional curvature on the image φ(M) is sufficiently small, then we are able to prove the harmonicity of the biharmonic map. It turns out that the fundamental tone of M is sufficiently large, then such a biharmonic map φ must be harmonic. In case where the map is an isometric immersion, we prove that if M satisfies a Sobolev inequality, then M must be minimal under the assumption that the Ln2-norm of the Ricci curvature on M is sufficiently small. Moreover it is shown that if the fundamental tone of a biharmonic submanifold is sufficiently big, then it is minimal.

Nonexistence of solutions to higher order evolution inequalities with nonlocal source term on Riemannian manifolds

We establish sufficient conditions for the nonexistence of nontrivial solutions to higher order evolution inequalities, with respect to the time variable. We consider a nonlocal source term, and work on complete noncompact Riemannian manifolds. The obtained conditions depend on the parameters of the problem and the geometry of the manifold. Our main result recovers some nonexistence theorems from the literature, established in the whole Euclidean space.

Gradient estimates for bounded solutions of semilinear elliptic equations and the Allen-Cahn equation on manifolds

This paper aims at deriving apriori bounds on the gradient of positve solutions to a class of semilinear elliptic equation, with applications focusing on establishing a liouville type property for the bounded solutions of the Allen-Cahn equation on a complete noncompact Riemannian manifold with nonnegative Ricci curvature.

Local Riesz Transform and Local Hardy Spaces on Riemannian Manifolds with Bounded Geometry

We prove that if \(\tau \) is a large positive number, then the atomic Goldberg-type space \({\mathfrak {h}}^1(N)\) and the space \({\mathfrak {h}}_{{\mathscr {R}}_\tau }^1(N)\) of all integrable functions on N of which local Riesz transform \({\mathscr {R}}_\tau \) is integrable, are the same space on any complete noncompact Riemannian manifold N with Ricci curvature bounded from below and positive injectivity radius. We also relate \({\mathfrak {h}}^1(N)\) to a space of harmonic functions on the slice \(N\times (0,\delta )\) for \(\delta >0\) small enough.

Gradient estimates of a nonlinear elliptic equation for the $V$-Laplacian on noncompact Riemannian manifolds

In this paper, we consider gradient estimates for positive solutions to the following equation $$\triangle_V u+au^p\log u=0$$ on complete noncompact Riemannian manifold with $k$-dimensional Bakry-Emery Ricci curvature bounded from below. Using the Bochner formula and the Cauchy inequality, we obtain upper bounds of $ \nabla u $ with respect to the lower bound of the Bakry-Emery Ricci curvature.

Instantaneous blow-up for nonlinear Sobolev type equations with potentials on Riemannian manifolds

<p style='text-indent:20px;'>We investigate Cauchy problems for two classes of nonlinear Sobolev type equations with potentials defined on complete noncompact Riemannian manifolds. The first one involves a polynomial nonlinearity and the second one involves a gradient nonlinearity. Namely, we derive sufficient conditions depending on the geometry of the manifold, the power nonlinearity, the behavior of the potential at infinity, and the initial data, for which the considered problems admit no nontrivial local weak solutions, i.e., an instantaneous blow-up occurs.</p>

Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free Argument

Abstract The concentration-compactness principle for the Trudinger–Moser-type inequality in the Euclidean space was established crucially relying on the Pólya–Szegő inequality which allows to adapt the symmetrization argument. As far as we know, the first concentration-compactness principle of Trudinger–Moser type in non-Euclidean settings, such as the Heisenberg (and more general stratified) groups where the Pólya–Szegő inequality fails, was found in [J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 2018, 3, Paper No. 84] by developing a nonsmooth truncation argument. In this paper, we establish the concentration-compactness principle of Trudinger–Moser type on any compact Riemannian manifolds as well as on the entire complete noncompact Riemannian manifolds with Ricci curvature lower bound. Our method is a symmetrization-free argument on Riemannian manifolds where the Pólya–Szegő inequality fails. This method also allows us to give a completely symmetrization-free argument on the entire Heisenberg (or stratified) groups which refines and improves a proof in the paper of Li, Lu and Zhu. Our results also show that the bounds for the suprema in the concentration-compactness principle on compact manifolds are continuous and monotone increasing with respect to the volume of the manifold.

A blow-up result for a nonlinear wave equation on manifolds: the critical case

We consider a inhomogeneous semilinear wave equation on a noncompact complete Riemannian manifold of dimension , without boundary. The reaction exhibits the combined effects of a critical term and of a forcing term. Using a rescaled test function argument together with appropriate estimates, we show that the equation admits no global solution. Moreover, in the special case when , our result improves the existing literature. Namely, our main result is valid without assuming that the initial values are compactly supported.

The sharp lower bound of the first Dirichlet eigenvalue for geodesic balls

On complete noncompact Riemannian manifolds with non-negative Ricci curvature, Li–Schoen proved the uniform Poincaré inequality for any geodesic ball. In this note, we obtain the sharp lower bound of the first Dirichlet eigenvalue of such geodesic balls, which implies the sharp Poincaré inequality for geodesic balls.

Critical and subcritical Trudinger-Moser inequalities on complete noncompact Riemannian manifolds

Though the sharp Trudinger-Moser inequalities on compact Riemannian manifolds have been known for some times, optimal constants for such inequalities on complete noncompact Riemannian manifolds are still unknown except in some special cases (such as hyperbolic spaces or Hadamard manifolds). In this paper, we will establish the sharp critical and subcritical Trudinger-Moser type inequalities with best constants on any complete and noncompact Riemannian manifold (M,g) whose Ricci curvature and its injectivity radius have lower bounds. (See Theorems 1.3 and 1.4.)More precisely, we will establish the following sharp critical Trudinger-Moser inequality on such manifolds (M,g,dVg):supu∈W1,n(M),||u||1,τ≤1⁡∫Mϕ(αn|u|nn−1)dVg≤C(n,τ), where τ>0, ϕ(t)=∑k=n−1∞tkk!, αn=nωn−11n−1, and ωn−1 is the area of the unit sphere in Rn, ||u||1,τ=(∫Mτ|u|n+|∇u|ndVg)1n. Moreover, we also prove that the following sharp subcritical Trudinger-Moser inequalitysupu∈W1,n(M),||∇u||n≤1⁡1||u||Ln(M)n∫Mϕ(α|u|nn−1)dVg≤C(n,α) in the sense that the supremum is finite for all α<αn, but infinite for α≥αn.The proofs of both the critical and subcritical Trudinger-Moser inequalities rely crucially on the Trudinger-Moser inequality on any compact manifold M with or without boundary, which is of its own interest. The important property we establish in this paper is that the upper bound C(n,M) for the Trudinger-Moser supremum is continuous and monotone increasing with respect to the volume of M. (See Theorems 1.1 and 1.2.) This upper bound property allows us to carry out a symmetrization-free argument from sharp local inequalities on compact manifolds to the optimal global inequalities on complete and noncompact Riemannian manifolds. Our optimal results sharpen substantially the existing ones on such Riemannian manifolds in the literature.

Compact Sobolev embeddings on non-compact manifolds via orbit expansions of isometry groups

Given a complete non-compact Riemannian manifold (M, g) with certain curvature restrictions, we introduce an expansion condition concerning a group of isometries G of (M, g) that characterizes the coerciveness of G in the sense of Skrzypczak and Tintarev (Arch Math 101(3): 259–268, 2013). Furthermore, under these conditions, compact Sobolev-type embeddings à la Berestycki-Lions are proved for the full range of admissible parameters (Sobolev, Moser-Trudinger and Morrey). We also consider the case of non-compact Randers-type Finsler manifolds with finite reversibility constant inheriting similar embedding properties as their Riemannian companions; sharpness of such constructions are shown by means of the Funk model. As an application, a quasilinear PDE on Randers spaces is studied by using the above compact embeddings and variational arguments.

Vanishing Theorems for Riemannian Manifolds with Nonnegative Scalar Curvature and Weighted p-Poincaré Inequality

In this paper, we give some vanishing theorems for harmonic p-forms on complete noncompact Riemannian manifolds satisfying a weighted p-Poincare inequality with nonnegative scalar curvature and under pointwise curvature pinching conditions which are bounded from above by the weight function.

Resolvent, heat kernel, and torsion under degeneration to fibered cusps

Manifolds with fibered cusps are a class of complete noncompact Riemannian manifolds including all locally symmetric spaces of rank one. We study the spectrum of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold undergoing degeneration to a manifold with fibered cusps. We obtain precise asymptotics for the resolvent, the heat kernel, and the determinant of the Laplacian. Using these asymptotics we obtain a topological description of the analytic torsion on a manifold with fibered cusps in terms of the R-torsion of the underlying manifold with boundary.

Liouville theorems for ancient caloric functions via optimal growth conditions

We prove two Liouville theorems for ancient nonnegative solutions of the heat equation on a complete noncompact Riemannian manifold with Ricci curvature bounded from below by $-K$, $K\geqslant 0$. If,

On the Hamilton's isoperimetric ratio in complete Riemannian manifolds of finite volume

We study a family of geometric variational functionals introduced by Hamilton, and considered later by Daskalopulos, Sesum, Del Pino and Hsu, in order to understand the behavior of maximal solutions of the Ricci flow both in compact and noncompact complete Riemannian manifolds of finite volume. The case of dimension two has some peculiarities, which force us to use different ideas from the corresponding higher-dimensional case. Under some natural restrictions, we investigate sufficient and necessary conditions which allow us to show the existence of connected regions with a connected complementary set (the so-called “separating regions”). In dimension higher than two, the associated problem of minimization is reduced to an auxiliary problem for the isoperimetric profile (with the corresponding investigation of the minimizers). This is possible via an argument of compactness in geometric measure theory valid for the case of complete finite volume manifolds. Moreover, we show that the minimum of the separating variational problem is achieved by an isoperimetric region. The dimension two requires different techniques of proof. The present results develop a definitive theory, which allows us to circumvent the shortening curve flow approach of the above mentioned authors at the cost of some applications of the geometric measure theory and of the Ascoli-Arzela's Theorem.

A maximum principle related to volume growth and applications

In this paper, we derive a new form of maximum principle for smooth functions on a complete noncompact Riemannian manifold M for which there exists a bounded vector field X such that $$\langle \nabla f,X\rangle \ge 0$$ on M and $$\rm {div} X\ge af$$ outside a suitable compact subset of M, for some constant $$a>0$$ , under the assumption that M has either polynomial or exponential volume growth. We then use it to obtain some Bernstein-type results for hypersurfaces immersed into a Riemannian manifold endowed with a Killing vector field, as well as to some results on the existence and size of minimal submanifolds immersed into a Riemannian manifold endowed with a conformal vector field.

Generalized Compactness for Finite Perimeter Sets and Applications to the Isoperimetric Problem

For a complete noncompact Riemannian manifold with bounded geometry, we prove a “generalized” compactness result for sequences of finite perimeter sets with uniformly bounded volume and perimeter in a larger space obtained by adding limit manifolds at infinity. We extend previous results contained in Nardulli (Asian J Math 18(1):1–28, 2014), in such a way that the main theorem is a generalization of the generalized existence theorem, i.e., Theorem 1 of Nardulli (Asian J Math 18(1):1–28, 2014). We replace C2,α locally asymptotic bounded geometry with C0 locally asymptotic bounded geometry.