Noncompact Riemannian Manifolds Research Articles

A singular Yamabe problem on manifolds with solid cones

Abstract We study the existence of conformal metrics on noncompact Riemannian manifolds with noncompact boundary, which are complete as metric spaces and have negative constant scalar curvature in the interior and negative constant mean curvature on the boundary. These metrics are constructed on smooth manifolds obtained by removing d-dimensional submanifolds from certain n-dimensional compact spaces locally modelled on generalized solid cones. We prove the existence of such metrics if and only if d > n - 2 2 {d>\frac{n-2}{2}} . Our main theorem is inspired by the classical results by Aviles–McOwen and Loewner–Nirenberg, known in the literature as the “singular Yamabe problem”.

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.

The Log-Sobolev Inequality for a Submanifold in Manifolds With Asymptotic Non-Negative Intermediate Ricci Curvature

We prove a sharp Log-Sobolev inequality for submanifolds of a complete non-compact Riemannian manifold with asymptotic non-negative intermediate Ricci curvature and Euclidean volume growth. Our work extends a result of Dong et al. (Acta Math. Sci. Ser. B (Engl. Ed.), 44(1):189–194 (2024)) which already generalizes Yi and Zheng (To appear in Chin. Ann. Math., (2023)) and Brendle (Comm. Pure Appl. Math. 75(3), 449–454 (2022)).

Symptotic behavior of solutions of the inhomogeneous Schrödinger equation on noncompact Riemannian manifolds

The paper studies the behavior of bounded solutions of the inhomogeneous Schrödinger equation on non-compact Riemannian manifolds under a variation of the right side of the equation. Various problems for homogeneous elliptic equations, in particular the Laplace-Beltrami equation and the stationary Schrödinger equation, have been considered by a number of Russian and foreign authors since the second half of the 20th century. In the first part of this paper, an approach to the formulation of boundary value problems based on the introduction of classes of equivalent functions will be developed. The relationship between the solvability of boundary value problems on an arbitrary non-compact Riemannian manifold with variation of inhomogeneity is also established. In the second part of the work, based on the results of the first part, properties of solutions of the inhomogeneous Schrödinger equation on quasi-model manifolds are investigated, and exact conditions for unique solvability of the Dirichlet problem and some other boundary value problems on these manifolds are found.

Existence of solutions to modified nonlinear Schrödinger equations on non-compact Riemannian manifolds

We study the existence of nontrivial solutions to a class of modified nonlinear Schrödinger equation on a complete non-compact N-dimensional (N≥3) Riemannian manifold with asymptotically non-negative Ricci curvature. Using the critical point theory, in a general Sobolev space instead of an Orlicz one, we prove the existence of a nontrivial non-negative solution to a class of modified Schrödinger equation with coercive potential. We also estimate the decay rate of the solution and show that the solution is exponentially decay.

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

Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds

AbstractThis note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with $$L^1$$ L 1 initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.

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.

Inclusions and noninclusions of Hardy type spaces on certain nondoubling manifolds

In this paper we establish inclusions and noninclusions between various Hardy type spaces on noncompact Riemannian manifolds M with Ricci curvature bounded from below, positive injectivity radius and spectral gap.Our first main result states that, if L is the positive Laplace–Beltrami operator on M, then the Riesz–Hardy space HR1(M) is the isomorphic image of the Goldberg type space h1(M) via the map L1/2(I+L)−1/2, a fact that is false in Rn. Specifically, HR1(M) agrees with the Hardy type space X1/2(M) recently introduced by the first three authors; as a consequence, we prove that HR1(M) does not admit an atomic characterisation.Noninclusions are mostly proved in the special case where the manifold is a Damek–Ricci space S. Our second main result states that HR1(S), the heat Hardy space HH1(S) and the Poisson–Hardy space HP1(S) are mutually distinct spaces, a fact which is in sharp contrast to the Euclidean case, where these three spaces agree.

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.

The fractional porous medium equation on noncompact Riemannian manifolds

The fractional porous medium equation on noncompact Riemannian manifolds

Stabilization of the critical semilinear wave equation on non-compact Riemannian manifold

We study the stability of the defocusing critical semilinear wave equation with locally distributed damping in an exterior domain. Under some assumptions on the Riemannian metric, the geometric multiplier technique and the compactness-uniqueness arguments were used to prove the unique continuation, the observability inequality respectively. Then the energy decay of the nonlinear system was obtained.

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.

Geodesic nets on non-compact Riemannian manifolds

Abstract A geodesic flower is a finite collection of geodesic loops based at the same point 𝑝 that satisfy the following balancing condition: the sum of all unit tangent vectors to all geodesic arcs meeting at 𝑝 is equal to the zero vector. In particular, a geodesic flower is a stationary geodesic net. We prove that, in every complete non-compact manifold with locally convex ends, there exists a non-trivial geodesic flower.

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.

Multiple solutions for Schrödinger equations on Riemannian manifolds via $$\nabla $$-theorems

We consider a smooth, complete and non-compact Riemannian manifold \((\mathcal {M},g)\) of dimension \(d \ge 3\), and we look for solutions to the semilinear elliptic equation $$\begin{aligned} -\varDelta _g w + V(\sigma ) w = \alpha (\sigma ) f(w) + \lambda w \quad \hbox {in }\mathcal {M}. \end{aligned}$$The potential \(V :\mathcal {M} \rightarrow \mathbb {R}\) is a continuous function which is coercive in a suitable sense, while the nonlinearity f has a subcritical growth in the sense of Sobolev embeddings. By means of \(\nabla \)-theorems introduced by Marino and Saccon, we prove that at least three non-trivial solutions exist as soon as the parameter \(\lambda \) is sufficiently close to an eigenvalue of the operator \(-\varDelta _g\).

Vanishing theorems for complete Riemannian manifolds with weighted p-Poincaré inequality

In this paper, we give some vanishing theorems for harmonic [Formula: see text]-forms on complete noncompact Riemannian manifolds satisfying a weighted [Formula: see text]-Poincaré inequality under the pointwise curvature pinching conditions which are bounded from above by the weight function. In some of our results, the condition on scalar curvature is no longer required as in previous ones.

МАССИВНЫЕ МНОЖЕСТВА, ПОРОЖДЁННЫЕ ПОЛУЛИНЕЙНЫМИ ЭЛЛИПТИЧЕСКИМИ ОПЕРАТОРАМИ НА НЕКОМПАКТНЫХ РИМАНОВЫХ МНОГООБРАЗИЯХ

One of the origins of the topic of this study is the classification theory of non-compact Riemannian surfaces. It is well known that on parabolic surfaces, any superharmonic functions bounded from below is the identical constant. Hyperbolic surfaces contain nontrivial superharmonic functions. This distinct property of parabolic surfaces form the basis for the definitions of parabolic manifolds with dimensions greater than two. The classification theory of Riemannian manifolds is directly related to Liouville-type theorems which assert the triviality of bounded solutions of elliptic equations. High efficiency in this topic was shown by the capacitive technique developed in the works of Grigoryan, Losev, Mazepa, and others. In particular, estimates were obtained for the dimensions of bounded harmonic functions and solutions of the stationary Schrödinger equation on noncompact Riemannian manifolds in terms of massive sets. In this paper, we study the properties of massive sets generated by a semilinear elliptic operator. It was possible to prove that the property of massiveness is preserved under variations of the potential. The current work generalizes or strengthens the results of Mazepa. A necessary condition for the existence of nontrivial bounded solutions of a semilinear equation is also obtained.