Search

Given a strongly local Dirichlet space and λ ⩾ 0 , we introduce a new notion of λ -subharmonicity for L loc 1 -functions, which we call local λ -shift defectivity, and which turns out to be equivalent to distributional λ -subharmonicity in the Riemannian case. We study the regularity of these functions on a new class of strongly local Dirichlet, so called locally smoothing spaces, which includes Riemannian manifolds (without any curvature assumptions), finite dimensional RCD spaces, Carnot groups, and Sierpinski gaskets. As a byproduct of this regularity theory, we obtain in this general framework a proof of a conjecture by Braverman, Milatovic, Shubin on the positivity of distributional L q -solutions of Δ f ⩽ f for complete Riemannian manifolds. [ABSTRACT FROM AUTHOR]

We obtain a vanishing result for solutions of the inequality |$|\Delta u| \leq q_{1} |u| + q_{2} |\nabla u|$| that decay to zero along a very general warped cylindrical end of a Riemannian manifold. The appropriate decay condition at infinity on |$u$| is related to the behavior of the potential functions |$q_{1}$| and |$q_{2}$| and to the asymptotic geometry of the end. The main ingredient is a new Carleman estimate of independent interest. Geometric applications to conformal deformations and to minimal graphs are presented. [ABSTRACT FROM AUTHOR]

On a complete Riemannian manifold (M, g), we consider L loc p distributional solutions of the differential inequality - Δ u + λ u ≥ 0 with λ > 0 a locally bounded function that may decay to 0 at infinity. Under suitable growth conditions on the L p norm of u over geodesic balls, we obtain that any such solution must be nonnegative. This is a kind of generalized L p -preservation property that can be read as a Liouville-type property for nonnegative subsolutiuons of the equation Δ u ≥ λ u . An application of the analytic results to L p growth estimates of the extrinsic distance of complete minimal submanifolds is also given. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the validity of first and second order Lp estimates for the solutions of the Poisson equation depending on the geometry of the underlying manifold. We first present Lp estimates of the gradient under the assumption that the Ricci tensor is lower bounded in a local integral sense, and construct the first counterexample showing that they are false, in general, without curvature restrictions. Next, we obtain Lp estimates for the second order Riesz transform (or, equivalently, the validity of Lp Calderón–Zygmund inequalities) on the whole scale 1 < p < ∞ by assuming that the injectivity radius is positive and that the Ricci tensor is either pointwise lower bounded, or non-negative in a global integral sense. When 1 < p ≤ 2, analogous Lp bounds on higher even order Riesz transforms are obtained provided that also the derivatives of Ricci are controlled up to a suitable order. [ABSTRACT FROM AUTHOR]

We show that two properly embedded self-shrinkers in Euclidean space that are sufficiently separated at infinity must intersect at a finite point. The proof is based on a localized version of the Reilly formula applied to a suitable ƒ-harmonic function with controlled gradient. In the immersed case, a new direct proof of the generalized half-space property is also presented. [ABSTRACT FROM AUTHOR]

On any complete Riemannian manifold M and for all p∈[2,∞), we prove a family of second-order Lp-interpolation inequalities that arise from the following simple Lp-estimate valid for every u∈C∞(M): ‖∇u‖pp≤‖uΔpu‖1∈[0,∞],where Δp denotes the p-Laplace operator. We show that these inequalities, in combination with abstract functional analytic arguments, allow to establish new global Sobolev regularity results for Lp-solutions of the Poisson equation for all p∈(1,∞), and new global Sobolev regularity results for the singular magnetic Schrödinger semigroups. [ABSTRACT FROM AUTHOR]

We give a self-contained treatment of the existence of a regular solution to the Dirichlet problem for harmonic maps into a geodesic ball on which the squared distance function from the origin is strictly convex. No curvature assumptions on the target are required. In this route we introduce a new deformation result which permits to glue a suitable Euclidean end to the geodesic ball without violating the convexity property of the distance function from the fixed origin. We also take the occasion to analyze the relationships between different notions of Sobolev maps when the target manifold is covered by a single normal coordinate chart. In particular, we provide full details on the equivalence between the notions of traced Sobolev classes of bounded maps defined intrinsically and in terms of Euclidean isometric embeddings. [ABSTRACT FROM AUTHOR]

Being motivated by the problem of deducing Lp-bounds on the second fundamental form of an isometric immersion from Lp-bounds on its mean curvature vector field, we prove a nonlinear Calderón-Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds. [ABSTRACT FROM AUTHOR]

We prove a $$\mathsf{C}^1$$ -elliptic estimate of the form   sup B (x, r / 4) | g r a d (ψ) | ≤ C min (1, r, (‖ S e c g ‖ L ∞ (B g (x, r)) + ϵ) − 1 / 2) { sup B (x, r / 2) | f | + sup B (x, r / 2) | ψ | }, valid on any complete Riemannian manifold $$M$$ and for any smooth solution of the Poisson equation $$\Delta \psi = f$$ which is defined in a neighbourhood of the geodesic ball $$B(x,r)$$. Above, $$C$$ is a constant which only depends on $$\dim(M)$$ and $$\epsilon >0$$ is arbitrary. In case of global solutions, the estimate is sensitive of the curvature growth on large balls and can be applied to deduce global results such as the zero-mean value property of $$f$$ as in the compact setting. [ABSTRACT FROM AUTHOR]

The paper aims at proving global height estimates for Killing graphs defined over a complete manifold with non-empty boundary. To this end, we first point out how the geometric analysis on a Killing graph is naturally related to a weighted manifold structure, where the weight is defined in terms of the length of the Killing vector field. According to this viewpoint, we introduce some potential theory on weighted manifolds with boundary and we prove a weighted volume estimate for intrinsic balls on the Killing graph. Finally, using these tools, we provide the desired estimate for the weighted height function in the assumption that the Killing graph has constant weighted mean curvature and the weighted geometry of the ambient space is suitably controlled. [ABSTRACT FROM AUTHOR]

In this paper we characterize the Neumann-parabolicity of manifolds with boundary in terms of a new form of the classical Ahlfors maximum principle and of a version of the so-called Kelvin--Nevanlinna--Royden criterion. The motivation underlying this study is to obtain new information on the geometry of graphs with prescribed mean curvature inside a Riemannian product of the type N x R. In this direction two kind of results will be presented: height estimates for constant mean curvature graphs parametrized over unbounded domains in a complete manifold, which extend results by A. Ros and H. Rosenberg valid for domains of R², and slice-type results for graphs whose superlevel sets have finite volume. Finally, the use of the Ahlfors maximum principle allows us to establish a connection between the Neumann-parabolicity and the Dirichlet-parabolicity commonly used in minimal surface theory. In particular, we will be able to give a deterministic proof of special cases of a result by R. Neel. [ABSTRACT FROM AUTHOR]

We prove that the existence of a Hurewicz fibration between certain spaces with the homotopy type of a CW-complex implies some topological restrictions on their universal coverings. This result is used to deduce differentiable and metric properties of maps between compact Riemannian manifolds under curvature restrictions. [ABSTRACT FROM AUTHOR]

In this paper we develop new geometric techniques to deal with the Dirichlet problem for a p-harmonic map from a compact manifold with boundary to a Cartan-Hadamard target manifold which is either 2-dimensional or rotationally symmetric. [ABSTRACT FROM AUTHOR]

In this paper we solve the relative homotopy Dirichlet problem for $$p$$ -harmonic maps from compact manifolds with boundary to compact manifolds of non-positive sectional curvature. The proof, which is based on the direct calculus of variations, uses some ideas of B. White to define the relative d-homotopy type of Sobolev maps. One of the main points of the proof consists in showing that the regularity theory by Hardt and Lin can be applied. A comprehensive uniqueness result for general complete targets with non-positive curvature is also given. [ABSTRACT FROM AUTHOR]

We study geometric properties of complete non-compact bounded self-shrinkers and obtain natural restrictions that force these hypersurfaces to be compact. Furthermore, we observe that, to a certain extent, complete self-shrinkers intersect transversally a hyperplane through the origin. When such an intersection is compact, we deduce spectral information on the natural drifted Laplacian associated to the self-shrinker. These results go in the direction of verifying the validity of a conjecture by H.D. Cao concerning the polynomial volume growth of complete self-shrinkers. A finite strong maximum principle in case the self-shrinker is confined into a cylindrical product is also presented. [ABSTRACT FROM AUTHOR]

A classical result by Alexander Grigor’yan states that on a stochastically complete manifold non-negative superharmonic L1-functions are necessarily constant. In this paper we construct explicit examples showing that, in the presence of an anisotropy of the space, the reverse implication does not hold. We also consider natural geometric situations where stochastically incomplete manifolds do not posses the above mentioned L1-Liouville property for superharmonic functions. [ABSTRACT FROM AUTHOR]

We survey some Lp-vanishing results for solutions of Bochner or Simons type equations with refined Kato inequalities, under spectral assumptions on the relevant Schrödinger operators. New aspects are included in the picture. In particular, an abstract version of a structure theorem for stable minimal hypersurfaces of finite total curvature is observed. Further geometric applications are discussed. [ABSTRACT FROM AUTHOR]

In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and L-Liouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scalar curvature estimates for gradient Ricci solitons under L conditions on the relevant quantities. [ABSTRACT FROM AUTHOR]

We prove global comparison results for the p-Laplacian on a p-parabolic manifold. These involve both real-valued and vector-valued maps with finite p-energy. Further L comparison principles in the non-parabolic setting are also discussed. [ABSTRACT FROM AUTHOR]