N-dimensional Complete Riemannian Manifold Research Articles

Liouville theorem for [formula omitted]-harmonic maps under non-negative [formula omitted]-Ricci curvature for non-positive [formula omitted

Let V be a C1-vector field on an n-dimensional complete Riemannian manifold (M,g). We prove a Liouville theorem for V-harmonic maps satisfying various growth conditions from complete Riemannian manifolds with non-negative (m,V)-Ricci curvature for m∈[−∞,0]∪[n,+∞] into Cartan–Hadamard manifolds, which extends Cheng’s Liouville theorem proved in Cheng (1980) for sublinear growth harmonic maps from complete Riemannian manifolds with non-negative Ricci curvature into Cartan–Hadamard manifolds. We also prove a Liouville theorem for V-harmonic maps from complete Riemannian manifolds with non-negative (m,V)-Ricci curvature for m∈[−∞,0]∪[n,+∞] into regular geodesic balls of Riemannian manifolds with positive upper sectional curvature bound, which extends the results of Hildebrandt et al. (1980) and Choi (1982). Our probabilistic proof of Liouville theorem for several growth V-harmonic maps into Hadamard manifolds enhances an incomplete argument in Stafford (1990). Our results extend the results due to Chen et al. (2012) and Qiu (2017) in the case of m=+∞ on the Liouville theorem for bounded V-harmonic maps from complete Riemannian manifolds with non-negative (∞,V)-Ricci curvature into regular geodesic balls of Riemannian manifolds with positive sectional curvature upper bound. Finally, we establish a connection between the Liouville property of V-harmonic maps and the recurrence property of ΔV-diffusion processes on manifolds. Our results are new even in the case V=∇f for f∈C2(M).

An overdetermined problem of the biharmonic operator on Riemannian manifolds

Let (M,g) be an n-dimensional complete Riemannian manifold with nonnegative Ricci curvature. In this paper, we consider an overdetermined problem of the biharmonic operator on a bounded smooth domain Ω in M. We deduce that the overdetermined problem has a solution only if Ω is isometric to a ball in mathbb{R}^{n}. Our method is based on using a P-function and the maximum principle argument. This result is a generalization of the overdetermined problem for the biharmonic equation in Euclidean space.

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.

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.

Curvature Estimates for Graphs Over Riemannian Domains

Let $$M^n$$ be a complete n-dimensional Riemannian manifold and $$\Gamma _f$$ the graph of a $$C^2$$ -function f defined on a metric ball of $$M^n$$ . In the same spirit of the estimates obtained by Heinz for the mean and Gaussian curvatures of a surface in $${\mathbb {R}}^3$$ which is a graph over an open disk in the plane, we obtain in this work upper estimates for $$\inf |R|$$ , $$\inf |A|$$ and $$\inf |H_k|$$ , where R, |A| and $$H_k$$ are, respectively, the scalar curvature, the norm of the second fundamental form and the k-th mean curvature of $$\Gamma _f$$ . From our estimates we obtain several results for graphs over complete manifolds. For example, we prove that if $$M^n,\;n\ge 3,$$ is a complete noncompact Riemannian manifold with sectional curvature bounded below by a constant c, and $$\Gamma _f$$ is a graph over M with Ricci curvature less than c, then $$\inf |A|\le 3(n-2)\sqrt{-c}$$ . This result generalizes and improves a theorem of Chern for entire graphs in $$\mathbb R^{n+1}$$ .

Global existence of smooth solutions for wave maps in de Sitter spacetime

In this paper, we study the problem of a harmonic map from de Sitter spacetime into a geodesic on an n-dimensional complete Riemannian manifold. We show that the global smooth solution exists for such wave map when the solution is a small perturbation along the geodesic solution by the method of weighted energy estimates.

A Liouville-type theorem for biharmonic maps between complete Riemannian manifolds with small energies

We prove a Liouville-type theorem for biharmonic maps from a complete Riemannian manifold of dimension n that has a lower bound on its Ricci curvature and positive injectivity radius into a Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the L^p-norm of the tension field is bounded and the n-energy of the map is sufficiently small, then every biharmonic map must be harmonic, where 2<p<n.

Li-Yau estimates for a semilinear parabolic equation on an evolving manifold

Let (M,g(t)) be a complete Riemannian manifold of dimension n, we obtain Li-Yau type gradient estimates on positive bounded solutions to the following semilinear parabolic equation&#x0D; ∂u(t,x) ∕ ∂t = Δ u(t,x) + a(x) us(t,x) -λu(t,x),&#x0D; where (t,x) ∈ ([0,T] × M), T &lt; ∞, s &gt; 1, λ ∈ ℝ and a ∈ C2(M) on evolving Riemannian metrics g(t) with bounded below Ricci tensor. The application of our gradient estimates yields the classical differential Harnack inequality, which compares a solution at some time with those at previous time.

A compactness theorem in Riemannian manifolds

In this paper, we use the m-Bakry–Emery Ricci tensor on a complete n-dimensional Riemannian manifold to obtain a compactness theorem including a diameter estimate. The proof is based on the Riccati comparison theorem.

On the exponent of convergence of negatively curved manifolds without Green’s function

In this paper we prove that for every complete n-dimensional Riemannian manifold without Green’s function and with its sectional curvatures satisfying K ≤−1, the exponent of convergence is greater than or equal to n − 1. Furthermore, we show that this inequality is sharp. This result is well known for manifolds with constant sectional curvatures K = −1.

Some L p Rigidity Results for Complete Manifolds with Harmonic Curvature

Let (M n , g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R m the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R m goes to zero uniformly at infinity if for $p\geq \frac n2$ , the L p -norm of R m is finite. Moreover, If R is positive, then (M n , g) is compact. As applications, we prove that (M n , g) is isometric to a spherical space form if for $p\geq \frac n2$ , R is positive and the L p -norm of R m is pinched in [0, C 1), where C 1 is an explicit positive constant depending only on n, p, R and the Yamabe constant. We give an isolation theorem of the trace-free Ricci curvature tensor of compact locally conformally flat Riemannian n-manifolds with constant positive scalar curvature, which extends Theorem 1 of Hebey and M. Vaugon (J. Geom. Anal. 6, 531–553, 1996). This result is sharp, and we can precisely characterize the case of equality. In particular, when n = 4, we recover results by Gursky (Indiana Univ. Math. J. 43, 747–774, 1994; Ann. Math. 148, 315–337, 1998).

Harnack differential inequalities for the parabolic equation u t = ℒF(u) on Riemannian manifolds and applications

In this paper, let (Mn, g) be an n-dimensional complete Riemannian manifold with the m-dimensional Bakry–Emery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation $${u_t} = LF\left( u \right) = \Delta F\left( u \right) - \nabla f \cdot \nabla F\left( u \right),$$ on compact Riemannian manifolds Mn, where F ∈ C2(0,∞), F′ > 0 and f is a C2-smooth function defined on Mn. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.

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.

Yau’s gradient estimates for a nonlinear elliptic equation

Let \({(M^n,g)}\) be an n-dimensional complete Riemannian manifold. We consider Yau’s gradient estimates for positive solutions to the following nonlinear equation $$\Delta u + au {\rm log} u=0$$ where a is a constant. As an application, we obtain the Liouville property for this equation in the case of a < 0. In addition, we illustrate, by giving concrete examples, that our results are sharp.

A new proof of a theorem of Petersen

Let M be an n-dimensional complete Riemannian manifold with Ricci curvature ⩾ n - 1. By developing some new techniques, Colding (1996) proved that the following three conditions are equivalent: 1) dGH(M,Sn) → 0; 2) the volume of M Vol(M) → Vol(Sn); 3) the radius of M rad(M) → π. By developing a different technique, Petersen (1999) gave the 4-th equivalent condition, namely he proved that the n + 1-th eigenvalue of M, λn+1(M) → n, is also equivalent to the radius of M, rad(M) → π, and hence the other two. In this paper, we use Colding’s techniques to give a new proof of Petersen’s theorem. We expect our estimates will have further applications.

Gradient estimates and Liouville type theorems for a nonlinear elliptic equation

Let (Mn, g) be an n-dimensional complete Riemannian manifold. We consider gradient estimates and Liouville type theorems for positive solutions to the following nonlinear elliptic equation: $$\Delta u+au\log u=0,$$ where a is a nonzero constant. In particular, for a < 0, we prove that any bounded positive solution of the above equation with a suitable condition for a with respect to the lower bound of Ricci curvature must be \({u\equiv 1}\). This generalizes a classical result of Yau.

Global existence of solutions to parabolic Monge–Ampère equations on Riemannian manifolds

In this paper, we study the Cauchy problem of the parabolic Monge–Ampère equation {ut−log{g−1det(gij+∇iju)}=−nloguinMn×(0,∞),u(x,0)=u0(x)inMn, where Mn is a compact complete Riemannian manifold of dimension n≥2, gij denotes the metric of Mn, g=det(gij)>0 and u0(x)>1 is a smooth function. We prove the global existence of solutions.

Théorèmes de type Obata pour certains feuilletages riemanniens

A theorem of Obata states that a complete Riemannian manifold of dimension n⩾2 admitting a nontrivial function f satisfying the differential equation Agradf=fIn is a sphere (Obata, 1962). In this paper, we propose to study the analogous situation for some Riemannian foliations of dimension 1 or codimension 1 on Riemannian manifolds of dimension n⩾3.

Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds

Let (M n , g) be an n-dimensional complete Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation: ut = � u + aulog u + bu on M n ×( 0, T ) ,w herea, b are two real constants. We derive local gradient estimates of the Li-Yau type for positive solutions of the above equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequali- ties are obtained. In particular, our results extend the ones of Davies in Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol 92, Cambridge University Press, Cambridge,1989, and Li and Xu in Adv Math 226:4456-4491 (2011).

Estimates for eigenvalues of a clamped plate problem on Riemannian manifolds

In this paper we study eigenvalues of a clamped plate problem on a bounded domain in an n-dimensional complete Riemannian manifold. By making use of Nash's theorem and introducing k free constants, we derive a universal bound for eigenvalues, which solves a problem proposed by Wang and Xia [16].