Complete Noncompact Riemannian Manifold Research Articles

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.

Gradient estimates for a nonlinear elliptic equation on complete noncompact Riemannian manifold

Gradient estimates for a nonlinear elliptic equation on complete noncompact Riemannian manifold

Rigidity of complete noncompact Riemannian manifolds with harmonic curvature

For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality involving $L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature and the Sobolev constant.

Solving Liouville-type Problems on Manifolds with Poincaré-Sobolev Inequality by Broadening q-Energy from Finite to Infinite

The aim of this article is to investigate Liouville-type problems on complete non-compact Riemannian manifolds with Poincaré-Sobolev Inequality. Two significant technical breakthroughs are demonstrated in research findings. The first breakthrough is an extension from non-flat manifolds with non-negative Ricci curvatures to curved manifolds with Ricci curvatures varying among negative values, zero, and positive values. Poincaré-Sobolev Inequality has been applied to overcome difficulties of an extension on manifolds. Poincaré-Sobolev Inequality has offered a special structure on curved manifolds with a mix of Ricci curvature signs. The second breakthrough is a generalization of $q$-energy from finite to infinite. At this point, a technique of $p$-balanced growth has been introduced to overcome difficulties of broadening from finite $q$-energy in $L^q$ spaces to infinite $q$-energy in non-$L^q$ spaces. An innovative computational method and new estimation techniques are illustrated. At the end of this article, Liouville-type results including vanishing properties for differential forms and constancy properties for differential maps have been verified on manifolds with Poincaré-Sobolev Inequality approaching to infinite $q$-energy growth.

A Note on “Riesz Transform for $$1 \le p \le 2$$ 1 ≤ p ≤ 2 Without Gaussian Heat Kernel Bound”

Consider the direct product manifold $$M_1 \times \cdots \times M_n$$ , where $$M_i$$ ( $$1 \le i \le n$$ ) are connected complete non-compact Riemannian manifolds satisfying the volume doubling property and Gaussian or sub-Gaussian estimates for the heat kernel. We obtain weak type (1, 1) (so $$L^p$$ -boundedness with $$1< p < 2$$ ) for the Riesz transform. As a consequence, we find that neither heat kernel Gaussian estimates nor sub-Gaussian estimates are necessary for weak (1, 1) property of Riesz transform.

Density of bounded maps in Sobolev spaces into complete manifolds

Given a complete noncompact Riemannian manifold $N^n$, we investigate whether the set of bounded Sobolev maps $(W^{1, p} \cap L^\infty) (Q^m; N^n)$ on the cube $Q^m$ is strongly dense in the Sobolev space $W^{1, p} (Q^m; N^n)$ for $1 \le p \le m$. The density always holds when $p$ is not an integer. When $p$ is an integer, the density can fail, and we prove that a quantitative trimming property is equivalent with the density. This new condition is ensured for example by a uniform Lipschitz geometry of $N^n$. As a byproduct, we give necessary and sufficient conditions for the strong density of the set of smooth maps $C^\infty (\overline{Q^m}; N^n)$ in $W^{1, p} (Q^m; N^n)$.

Gradient estimates for porous medium equations under the Ricci flow

A local gradient estimate for positive solutions of porous medium equations on complete noncompact Riemannian manifolds under the Ricci flow is derived. Moreover, a global gradient estimate for such equations on compact Riemannian manifolds is also obtained.

Toeplitz Operators and the Roe-Higson Type Index Theorem in Riemannian Surfaces

Let $M$ be a non-compact complete Riemannian manifold of dimension two and $N$ a circle in $M$. We assume that $M$ is partitioned by $N$. We define a unital $C^{\ast}$-algebra $C_{b}^{\ast}(M)$, which is slightly larger than the Roe algebra of $M$. We also construct $[u_{\phi}]$ in $K_{1}(C_{b}^{\ast}(M))$, which is a counter part of Roe's odd index class. We prove that Connes' pairing of Roe's cyclic one-cocycle with $[u_{\phi}]$ is equal to the Fredholm index of a Toeplitz operator on $N$. It is a part of an extension of the Roe-Higson index theorem to even-dimensional partitioned manifolds.

Stable solutions of the Yamabe equation on non-compact manifolds

We consider the Yamabe equation on a complete non-compact Riemannian manifold and study the condition of stability of solutions. If $(M^m ,g)$ is a closed manifold of constant positive scalar curvature, which we normalize to be $m(m-1)$, we consider the Riemannian product with the $n$-dimensional Euclidean space: $(M^m \times \mathbb{R}^n , g+ g_E )$. And we study, as in [2], the solution of the Yamabe equation which depends only on the Euclidean factor. We show that there exists a constant $\lambda (m,n)$ such that this solution is stable if and only if $\lambda_1 \geq \lambda (m,n)$, where $\lambda_1$ is the first positive eigenvalue of $-\Delta_g$. We compute $\lambda (m,n)$ numerically for small values of $m,n$ showing in these cases that the Euclidean minimizer is stable in the case $M=S^m$ with the metric of constant curvature. This implies that the same is true for any closed manifold with a Yamabe metric.

Harmonic transforms of complete Riemannian manifolds

Vanishing theorems for harmonic and infinitesimal harmonic transformations of complete Riemannian manifolds are proved. The proof uses well-known Liouville theorems on subharmonic functions on noncompact complete Riemannian manifolds.

Hamilton's gradient estimates for fast diffusion equations under the Ricci flow

Let M be a complete noncompact Riemannian manifold of dimension n. We derive a Hamilton's gradient estimate for positive solutions of the fast diffusion equations∂u∂t=Δum,1−4n+8<m<1 on M×(−∞,0] under the Ricci flow.

Eigenvalues of the drifting Laplacian on complete noncompact Riemannian manifolds

In this paper, we investigate eigenvalues of the eigenvalue problem with Dirichlet boundary condition of the drifting Laplacian on an n-dimensional, complete noncompact Riemannian manifold. Some estimates for eigenvalues are obtained. By utilizing Cheng and Yang recursion formula, we give a sharp upper bound of the kth eigenvalue. As we know, product Riemannian manifolds, Ricci solitons and self-shrinkers are some important Riemannian manifolds. Therefore, we investigate the eigenvalues of the drifting Laplacian on those Riemannian manifolds. In particular, by some theorems of classification for Ricci solitons, we can obtain some eigenvalue inequalities of drifting Laplacian on the Ricci solitons with certain conditions.

Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds

We are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact Riemannian manifolds. In particular, we highlight the interplay between the geometry of the underlying manifold, the power nonlinearity and the behavior of the potential at infinity.

The topology of an open manifold with radial curvature bounded from below by a model surface with finite total curvature and examples of model surfaces

We construct distinctive surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we prove that a complete noncompact Riemannian manifold M is homeomorphic to the interior of a compact manifold with boundary if the manifold M is not less curved than a noncompact model surface of revolution and if the total curvature of the model surface is finite and less than 2π. By the first result mentioned above, the second result covers a much wider class of manifolds than that of complete noncompact Riemannian manifolds whose sectional curvatures are bounded from below by a constant.

Gradient estimates of Li Yau type for a general heat equation on Riemannian manifolds

In this paper, we consider gradient estimates on complete noncompact Riemannian manifolds $(M,g)$ for the following general heat equation \[ u_t=\Delta _V u+au\log u+bu \] where $a$ is a constant and $b$ is a differentiable function defined on $M\times [0, \infty )$. We suppose that the Bakry-Emery curvature and the $N$-dimensional Bakry-Emery curvature are bounded from below, respectively. Then we obtain the gradient estimate of Li-Yau type for the above general heat equation. Our results generalize the work of Huang-Ma ([4]) and Y. Li ([6]), recently.

Riesz transforms of the Hodge-de Rham Laplacian on Riemannian manifolds

Let $M$ be a complete non-compact Riemannian manifold satisfying the doubling volume property. Let $\overrightarrow{\Delta}$ be the Hodge-de Rham Laplacian acting on $1$-differential forms. According to the Bochner formula, $\overrightarrow{\Delta}=\nabla^*\nabla+R_+-R_-$ where $R_+$ and $R_-$ are respectively the positive and negative part of the Ricci curvature and $\nabla$ is the Levi-Civita connection. We study the boundedness of the Riesz transform $d^*(\overrightarrow{\Delta})^{-\frac{1}{2}}$ from $L^p(\Lambda^1T^*M)$ to $L^p(M)$ and of the Riesz transform $d(\overrightarrow{\Delta})^{-\frac{1}{2}}$ from $L^p(\Lambda^1T^*M)$ to $L^p(\Lambda^2T^*M)$. We prove that, if the heat kernel on functions $p_t(x,y)$ satisfies a Gaussian upper bound and if the negative part $R_-$ of the Ricci curvature is $\epsilon$-sub-critical for some $\epsilon\in[0,1)$, then $d^*(\overrightarrow{\Delta})^{-\frac{1}{2}}$ is bounded from $L^p(\Lambda^1T^*M)$ to $L^p(M)$ and $d(\overrightarrow{\Delta})^{-\frac{1}{2}}$ is bounded from $L^p(\Lambda^1T^*M)$ to $L^p(\Lambda^2T^* M)$ for $p\in(p_0',2]$ where $p_0>2$ depends on $\epsilon$ and on a constant appearing in the doubling volume property. A duality argument gives the boundedness of the Riesz transform $d(\Delta)^{-\frac{1}{2}}$ from $L^p(M)$ to $L^p(\Lambda^1T^*M)$ for $p\in [2,p_0)$ where $\Delta$ is the non-negative Laplace-Beltrami operator. We also give a condition on $R_-$ to be $\epsilon$-sub-critical under both analytic and geometric assumptions.

Weighted Hardy and Rellich type inequalities on Riemannian manifolds

In this paper we present new results on two-weight Hardy, Hardy–Poincare and Rellich type inequalities with remainder terms on a complete noncompact Riemannian Manifold M. The method we use is flexible enough to obtain more weighted Hardy type inequalities. Our results improve and include many previously known results as special cases.

Gradient estimates for a nonlinear diffusion equation on complete manifolds

This paper deals with the gradient estimates of the Hamilton type for the positive solutions to the following nonlinear diffusion equation: $${u_t} = \Delta u + \nabla \phi \cdot \nabla u + a\left( x \right)u\ln u + b\left( x \right)u$$ on a complete noncompact Riemannian manifold with a Bakry-Emery Ricci curvature bounded below by −K (K ≥ 0), where φ is a C2 function, a(x) and b(x) are C1 functions with certain conditions.

Sobolev spaces with variable exponents on complete manifolds

We study variable exponent function spaces on complete non-compact Riemannian manifolds. Using classic assumptions on the geometry, continuous embeddings between Sobolev and Hölder function spaces are obtained. We prove compact embeddings of H -invariant Sobolev spaces, where H is a compact Lie subgroup of the group of isometries of the manifold. As an application, we prove the existence of non-trivial solutions to non-homogeneous q ( x ) -Laplace equations.

Gap theorems for locally conformally flat manifolds

In this paper, we prove a gap result for a locally conformally flat complete non-compact Riemannian manifold with bounded non-negative Ricci curvature and a scalar curvature average condition. We show that if it has positive Green function, then it is flat. This result is proved by setting up new global Yamabe flow. Other extensions related to bounded positive solutions to a Schrodinger equation are also discussed. A global existence of Yamabe flow on a locally conformally flat complete non-compact Riemannian manifold with bounded non-negative sectional curvature is proved.