Complete Noncompact Riemannian Manifold Research Articles

Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian

We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we construct such manifolds with dense embedded point spectrum and sharp curvature bounds.

A maximum principle at infinity with applications to geometric vector fields

We derive a new form of maximum principle, applicable to a vector field with nonnegative divergence in a connected, oriented, complete noncompact Riemannian manifold. We then use it to obtain some applications to Killing vector fields. More precisely, we first show that, under a reasonable condition at infinity, an orientable, connected, complete noncompact hypersurface of a Riemannian manifold, transversal to a Killing vector field of constant norm and with nonnegative second fundamental form, is totally geodesic. We also deal with the case of a hypersurface of constant mean curvature – instead of nonnegative second fundamental form, and show that it has to be totally geodesic too.

Yamabe flow and metrics of constant scalar curvature on a complete manifold

In this paper, we give a condition on the initial metric which makes the global existence of Yamabe flow and we study the global behavior of the Yamabe flow in a complete noncompact Riemannian manifold. As application we study the ADM mass monotonicity of Yamabe flow in AF manifolds. We use the variational method to study the existence problem of metrics with constant scalar curvature on complete non-compact Riemannian manifolds and we can give a partial affirmative answer to a question posed by Kazdan (Math Ann 261(2):227–234, 1982).

Local Hölder continuity of the isoperimetric profile in complete noncompact Riemannian manifolds with bounded geometry

For a complete noncompact connected Riemannian manifold with bounded geometry $M^n$, we prove that the isoperimetric profile function $I_{M^n}$ is a locally $\left(1-\frac{1}{n}\right)$-H\"older continuous function and so in particular it is continuous. Here for bounded geometry we mean that $M$ have $Ricci$ curvature bounded below and volume of balls of radius $1$, uniformly bounded below with respect to its centers. We prove also the equivalence of the weak and strong formulation of the isoperimetric profile function in complete Riemannian manifolds which is based on a lemma having its own interest about the approximation of finite perimeter sets with finite volume by open bounded with smooth boundary ones of the same volume. Finally the upper semicontinuity of the isoperimetric profile for every metric (not necessarily complete) is shown.

Rigidity of vector valued harmonic maps of linear growth

Consider vector valued harmonic maps of at most linear growth, defined on a complete non-compact Riemannian manifold with non-negative Ricci curvature. For the square of the Jacobian of such maps, we report a strong maximum principle, and equalities among its supremum, its asymptotic average, and its large-time heat evolution.

Hamilton’s gradient estimate for fast diffusion equations under geometric flow

Suppose that \lt i \gt M \lt /i \gt is a complete noncompact Riemannian manifold of dimension \lt i \gt n \lt /i \gt . In the present paper, we obtain a Hamilton's gradient estimate for positive solutions of the fast diffusion equations $ \dfrac{\partial u}{\partial t} = \Delta u^m ,\qquad 1-\dfrac{4}{n+8} \lt m \lt 1 $ on $M\times (-\infty, 0]$ under the geometric flow.

Sharp Growth Estimates for Warping Functions in Multiply Warped Product Manifolds

By applying an average method in PDE, we obtain a dichotomy between "constancy" and "infinity" of the warping functions on complete noncompact Riemannian manifolds for an appropriate isometric immersion of a multiply warped product manifold $N_1\times_{f_2} N_2 \times \cdots \times _{f_k} N_k\, $ into a Riemannian manifold. Generalizing the earlier work of the authors in [{Glasg. Math. J. 51 (2009) 579-592], we establish sharp inequalities between the mean curvature of the immersion and the sectional curvatures of the ambient manifold under the influence of quantities of a purely analytic nature (the growth of the warping functions). Several applications of our growth estimates are also presented.

Prescribed k-Curvature Problems on Complete Noncompact Riemannian Manifolds

Abstract To study the prescribed $k$-curvature problem of 2nd-order symmetric curvature tensors on complete noncompact Riemannian manifolds, we consider a class of fully nonlinear elliptic partial differential equations. It is proved that on a Riemannian manifold with negative sectional curvature and Ricci curvature bounded from below, the equation is solvable provided that all the eigenvalues of the tensor are negative. The result is applicable to the prescribed $k$-curvature problems of modified Schouten tensor and Bakry–Émery Ricci tensor.

Gradient estimates for the Allen-Cahn equation on Riemannian manifolds

In this paper, we consider bounded positive solutions to the Allen-Cahn equation on complete noncompact Riemannian manifolds without boundary. We derive gradient estimates for those solutions. As an application, we get a Liouville type theorem on manifolds with nonnegative Ricci curvature.

Elliptic problems on complete non-compact Riemannian manifolds with asymptotically non-negative Ricci curvature

In this paper we discuss the existence and non-existence of weak solutions to parametric equations involving the Laplace–Beltrami operator Δg in a complete non-compact d-dimensional (d≥3) Riemannian manifold (M,g) with asymptotically non-negative Ricci curvature and intrinsic metric dg. Namely, our simple model is the following problem −Δgw+V(σ)w=λα(σ)f(w)in Mw≥0in Mwhere V is a positive coercive potential, α is a positive bounded function, λ is a real parameter and f is a suitable continuous nonlinear term. The existence of at least two non-trivial bounded weak solutions is established for large value of the parameter λ requiring that the nonlinear term f is non-trivial, continuous, superlinear at zero and sublinear at infinity. Our approach is based on variational methods. No assumptions on the sectional curvature, as well as symmetry theoretical arguments, are requested in our approach.

Some Weighted Norm Inequalities on Manifolds

Let M be a complete non-compact Riemannian manifold satisfying the volume doubling property and the Gaussian upper bounds. Denote by Δ the Laplace-Beltrami operator and by ∇ the Riemannian gradient. In this paper, the author proves the weighted reverse inequality \(\left\| {{\Delta ^{\frac{1}{2}}}f} \right\|_{L^p(w)}\leq C\left\| {|\nabla f|} \right\|_{L^p(w)}\), for some range of p determined by M and w. Moreover, a weak type estimate is proved when p = 1. Some weighted vector-valued inequalities are also established.

Gaussian heat kernel estimates: From functions to forms

Abstract On a complete non-compact Riemannian manifold satisfying the volume doubling property, we give conditions on the negative part of the Ricci curvature that ensure that, unless there are harmonic 1-forms, the Gaussian heat kernel upper estimate on functions transfers to one-forms. These conditions do no entail any constraint on the size of the Ricci curvature, only on its decay at infinity.

On Manifolds with Lower Quadratic Curvature Decay and Volume Pinching

Abstract We give some conditions for a complete noncompact Riemannian manifold with lower quadratic curvature decay to have finite topological type. Most of our curvature conditions are much weaker than the assumptions in Lott [5] and in Sha–Shen [10], hence our results can be viewed as natural generalizations of those works.

A vanishing result for the supersymmetric nonlinear sigma model in higher dimensions

We prove a vanishing result for critical points of the supersymmetric nonlinear sigma model on complete non-compact Riemannian manifolds of positive Ricci curvature that admit an Euclidean type Sobolev inequality, assuming that the dimension of the domain is bigger than two and that a certain energy is sufficiently small.

Sharp Isoperimetric Inequalities for Small Volumes in Complete Noncompact Riemannian Manifolds of Bounded Geometry Involving the Scalar Curvature

Abstract We provide an isoperimetric comparison theorem for small volumes in an $n$-dimensional Riemannian manifold $(M^n,g)$ with $C^3$ bounded geometry in a suitable sense involving the scalar curvature function. Under $C^3$ bounds of the geometry, if the supremum of scalar curvature function $S_g<n(n-1)k_0$ for some $k_0\in \mathbb{R}$, then for small volumes the isoperimetric profile of $(M^n,g)$ is less then or equal to the isoperimetric profile of the complete simply connected space form of constant sectional curvature $k_0$. This work generalizes Theorem $2$ of [12] in which the same result was proved in the case where $(M^n, g)$ is assumed to be compact. As a consequence of our result we give an asymptotic expansion in Puiseux series up to the 2nd nontrivial term of the isoperimetric profile function for small volumes, generalizing our earlier asymptotic expansion [29]. Finally, as a corollary of our isoperimetric comparison result, it is shown that for small volumes the Aubin–Cartan–Hadamard’s conjecture is true in any dimension $n$ in the special case of manifolds with $C^3$ bounded geometry, and $S_g<n(n-1)k_0$. Two different intrinsic proofs of the fact that an isoperimetric region of small volume is of small diameter. The 1st under the assumption of mild bounded geometry, that is, positive injectivity radius and Ricci curvature bounded below. The 2nd assuming the existence of an upper bound of the sectional curvature, positive injectivity radius, and a lower bound of the Ricci curvature.

Elliptic gradient estimates for the doubly nonlinear diffusion equation

In this paper we establish elliptic type gradient estimates for positive solutions to the doubly nonlinear diffusion equation on compact or complete noncompact Riemannian manifolds. As special cases, we can derive elliptic type gradient estimates for positive solutions to the p-Laplace parabolic equations and the fast diffusion equations. Harnack inequalities derived from these elliptic type gradient estimates can be used to compare solutions at the same time.

Gradient estimates and Harnack inequalities for Yamabe-type parabolic equations on Riemannian manifolds

Let (Mn,g) be a complete noncompact n-dimensional Riemannian manifolds. In this paper, we consider the following Yamabe-type parabolic equationut=Δu+au+buα on Mn×[0,∞). We give a global gradient estimate of Hamilton-type for positive smooth solutions of this equation provided that Ricci curvature bounded from below. As its application, we show a dimension-free Harnack inequality and a Liouville-type theorem for nonlinear elliptic equations.

Growth of the Eigensolutions of Laplacians on Riemannian Manifolds I: Construction of Energy Function

Abstract In this paper, we consider the eigensolutions of $-\Delta u+ Vu=\lambda u$, where $\Delta $ is the Laplacian on a non-compact complete Riemannian manifold. We develop Kato’s methods on manifold and establish the growth of the eigensolutions as r goes to infinity based on the asymptotical behaviors of $\Delta r$ and V (x), where r = r(x) is the distance function on the manifold. As applications, we prove several criteria of absence of eigenvalues of Laplacian, including a new proof of the absence of eigenvalues embedded into the essential spectra of free Laplacian if the radial curvature of the manifold satisfies $ K_{\textrm{rad}}(r)= -1+\frac{o(1)}{r}$.

A note on Anosov flows of non-compact Riemannian manifolds

In this note we formulate a condition for complete non-compact Riemannian manifolds, which implies no conjugate points in case that the geodesic flow is Anosov with respect to the Sasaki metric.

On a class of nonhomogeneous elliptic equations on noncompact Riemannian manifolds

ABSTRACTLet (M, g) be a complete noncompact Riemannian manifold . In this paper, a fixed point result and a version of the Trudinger–Moser inequality are employed to establish sufficient conditions for the existence of solutions for quasilinear nonhomogeneous elliptic equation of the type Here the function V(x) can change sign, f(x, u) is possibly discontinuous and can enjoy exponential critical growth and h belongs to the dual of an appropriated function space.