P-Parabolicity and a Generalized Bochner’s Method with Applications

We establish existence and uniqueness theorems for V-harmonic maps from complete noncompact manifolds. This class of maps includes Hermitian harmonic maps, Weyl harmonic maps, affine harmonic maps, and Finsler harmonic maps from a Finsler manifold into a Riemannian manifold. We also obtain a Liouville type theorem for V-harmonic maps. In addition, we prove a V-Laplacian comparison theorem under the Bakry-Emery Ricci condition.

In this paper, we study complete noncompact pseudohermitian manifolds with positive spectrum of the sub-Laplacian. We prove splitting-type theorems for a class of complete noncompact pseudohermitian manifolds with vanishing torsion whose spectrum of the sub-Laplacian has an optimal positive lower bound. These can be viewed as the CR analogue of theorems of Li–Wang and the equality case of a theorem of Cheng.

The object of study in this paper is the Ax-operator associated to a Killing field (resp. to an infinitesimal analytic transformation) on Riemannian (resp. Kahler) complete but noncompact manifolds. We recall (§ 1) Kostant's and Lichneroivicz's Theorems for that operator in the compact case and we dedicate, § 2 and § 3 to study all the simply-connected noncompact manifolds in which similar results to that theorems do not hold, we give examples of all these cases. § 4 is dedicated to give sufficient conditions in order to ensure the validity of similar results to that above mentioned, on complete noncompact manifolds. The hypothesis used refer to vector fields whose norm are either pointwise bounded or globally bounded.

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.

On the essential spectrum of complete non-compact manifolds

Generalized Ricci flow II: Existence for complete noncompact manifolds

Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = ( x 1 , ... , x N ) ∈ M N for Lebesgue measure in M N , the measure has a unique p –mean e p ( x ). As a consequence, if X = ( X 1 , ... , X N ) is a M N -valued random variable with absolutely continuous law, then almost surely μ ( X ( ω )) has a unique p –mean. In particular if ( X n ) n ≥ 1 is an independent sample of an absolutely continuous law in M , then the process e p,n ( ω ) = e p ( X 1 ( ω ), ... , X n ( ω )) is well-defined. Assume M is compact and consider a probability measure ν in M . Using partial simulated annealing, we define a continuous semimartingale which converges in probability to the set of minimizers of the integral of distance at power p with respect to ν . When the set is a singleton, it converges to the p –mean.

In this paper, we consider $p$-harmonic functions on complete Riemannian manifolds and give different proofs of some main theorems by Chang-Chen-Wei, in [3]. Moreover, we are able to refine their results in case of weakly $p$-harmonic functions. Some applications to study the connectedness at infinity of stable minimal hypersurfaces are also given.

On a complete noncompact Kähler manifold we prove that the bottom of the spectrum for the Laplacian is bounded from above by m2 if the Ricci curvature is bounded from below by −2(m+1). Then we show that if this upper bound is achieved then either the manifold is connected at infinity or it has two ends and in this case it is diffeomorphic to the product of the real line with a compact manifold and we determine the metric.

In this paper, under m-Bakry–Emery (or $$\infty $$ -Bakry–Emery) pseudohermitian Ricci curvature condition, we derive subgradient estimate for positive solutions of $$\Delta _{H,\phi } u(x)=-\lambda u(x)$$ on a weighted complete noncompact pseudohermitian manifold which satisfies the sub-Laplacian comparison property.

Some regularity results for [formula omitted]-harmonic mappings between Riemannian manifolds

We extend a Yamabe-type invariant of the Dirac operator to noncompact manifolds and show that as in the compact case this invariant is bounded by the corresponding invariant of the standard sphere. Further, this invariant will lead to an obstruction of the conformal compactification of complete noncompact manifolds.

Let M ∘ M^\circ be a complete noncompact manifold and g g an asymptotically conic Riemaniann metric on M ∘ M^\circ , in the sense that M ∘ M^\circ compactifies to a manifold with boundary M M in such a way that g g becomes a scattering metric on M M . Let Δ \Delta be the positive Laplacian associated to g g , and P = Δ + V P = \Delta + V , where V V is a potential function obeying certain conditions. We analyze the asymptotics of the spectral measure d E ( λ ) = ( λ / π i ) ( R ( λ + i 0 ) − R ( λ − i 0 ) ) dE(\lambda ) = (\lambda /\pi i) \big ( R(\lambda +i0) - R(\lambda - i0) \big ) of P + 1 / 2 P_+^{1/2} , where R ( λ ) = ( P − λ 2 ) − 1 R(\lambda ) = (P - \lambda ^2)^{-1} , as λ → 0 \lambda \to 0 , in a manner similar to that done by the second author and Vasy (2001) and by the first two authors (2008, 2009). The main result is that the spectral measure has a simple, ‘conormal-Legendrian’ singularity structure on a space which was introduced in the 2008 work of the first two authors and is obtained from M 2 × [ 0 , λ 0 ) M^2 \times [0, \lambda _0) by blowing up a certain number of boundary faces. We use this to deduce results about the asymptotics of the wave solution operators cos ⁡ ( t P + ) \cos (t \sqrt {P_+}) and sin ⁡ ( t P + ) / P + \sin (t \sqrt {P_+})/\sqrt {P_+} , and the Schrödinger propagator e i t P + e^{itP_+} , as t → ∞ t \to \infty . In particular, we prove the analogue of Price’s law for odd-dimensional asymptotically conic manifolds. In future articles, this result on the spectral measure will be used to (i) prove restriction and spectral multiplier estimates on asymptotically conic manifolds, and (ii) prove long-time dispersion and Strichartz estimates for solutions of the Schrödinger equation on M M , provided M M is nontrapping.

In this paper we will prove a maximum principle for the solutions of linear parabolic equation on complete non-compact manifolds with a time varying metric. We will prove the convergence of the Neumann Green function of the conjugate heat equation for the Ricci flow in $B_k\times (0,T)$ to the minimal fundamental solution of the conjugate heat equation as $k\to\infty$. We will prove the uniqueness of the fundamental solution under some exponential decay assumption on the fundamental solution. We will also give a detail proof of the convergence of the fundamental solutions of the conjugate heat equation for a sequence of pointed Ricci flow $(M_k\times (-\alpha,0],x_k,g_k)$ to the fundamental solution of the limit manifold as $k\to\infty$ which was used without proof by Perelman in his proof of the pseudolocality theorem for Ricci flow [P].

We present some new Stokes' type theorems on complete non-compact manifolds that extend, in different directions, previous work by Gaffney and Karp and also the so called Kelvin-Nevanlinna-Royden criterion for (p-)parabolicity. Applications to comparison and uniqueness results involving the p-Laplacian are deduced.