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In this paper we explore Liouville's theorem on Riemannian cones as defined below. We also study the Strong Liouville Property, that is, the property of a cone having spaces of harmonic functions of a fixed polynomial growth of finite dimension., Comment: This is a first version. Criticisms are welcomed

We discuss the behavior of harmonic functions on Riemannian cones as defined below and Lioville's theorem., Comment: Version submitted to Analysis and PDE in Latin America - ICMAM 2022 Latin America

We study the subsequential convergence of singular solutions to the Ricci flow with prescribed constant in space geodesic curvature on compact surfaces with boundary. Furthermore, we show that in the particular case of rotational symmetry, this convergence does not depend on the sign of the geodesic curvature of the boundary.

We show the existence of a positive solution to certain weakly coupled elliptic systems of sublinear growth and homogenous Dirichlet boundary conditions. Our result generalizes known results in the case of sublinear systems of two weakly coupled equations, as, for instance Lane-Emden systems and not only applies to bounded but to thin unbounded domains (that is, unbounded domains contained in a region bounded by two parallel hyperplanes)., Comment: Criticisms are quite welcome

In this short paper we show that March's criterion for the existence of a bounded non constant harmonic function on a weak model is also a necessary and sufficient condition for the solvability of the Dirichlet problem at infinity on a slight generalisation of a weak model (rotationally symmetric) metric on $\mathbb{R}^n$., Comment: Comments and criticisms are welcome

In this paper we prove that in a three-manifold with finitely many expansive ends, such that each end has a neighborhood where the curvature is bounded above by a negative constant, the Dirichlet problem at infinity is solvable, and hence that such manifolds posses a wealth of bounded non constant harmonic functions (and thus, Liouville's theorem does not hold). In the case of infinitely many expansive ends, we show that if each end has a neighborhood where the curvature is bounded above by a negative constant, then the Dirichlet problem at infinity is solvable for continuous boundary data at infinity which is uniformly bounded. Our method is based on a result that does not require explicit curvature assumptions, and hence it can be applied to other situations: we present an example of a metric on an end with curvature of indefinite sign (no matter how long we go along the end) for which the Dirichlet Problem at Infinity is solvable with respect to that end. We also present a related result in the case of surfaces with a pole which generalises a celebrated criteria of Milnor., Comment: Comments and criticisms more than welcome. With respect to the previous version, we have added a result on two dimensions

In this short paper we show a sufficient condition for the solvability of the Dirichlet problem at infinity in Riemannian cones (as defined below).This condition is related to a celebrated result of Milnor that classifies parabolic surfaces. When applied tosmooth Riemannian manifolds with a special type of metrics (which generalise rotational symmetry) we obtain generalisations of classical criteria for the solvability of the Dirichlet problem at infinity. Our proof is short and elementary: it uses separation of variables and comparison arguments for ODE's., Comment: Criticisms are more than welcome

In this paper we give a proof via the contraction mapping principle of a Bloch-type theorem for normalised Bochner-Takahashi $K$-mappings, which are solutions to equations of the form $Lu=0$, where $L$ is the heat operator., Comment: Every criticism is welcome. The ideas presented here are a refinement (and an improvement) of those presented in "On Bloch's theorem and the contraction mapping principle" arxiv:1702.01080v3

In this paper we prove a general stability result for higher order geometric flows on the circle, which basically states that if the initial condition is close to a round circle, the curve evolves smoothly and exponentially fast towards a circle, and we improve on known convergence rates (which we believe are almost sharp). The polyharmonic flow is an instance of the flows to which our result can be applied (and of course we present some other examples)., Comment: Comments and criticisms are welcome

In this paper we give rates of convergence for the $p$-curve shortening flow for $p\geq 1$ an integer, which improves on the known estimates and which are probably sharp., Comment: Any, comments and suggestions are more than welcome

In this paper we give a new proof, relying on Banach's contraction mapping principle, of a celebrated theorem of Andr\'e Bloch. Also, via the same contraction mapping principle, we give a proof of a Bloch type theorem for normalised Wu $K$-mappings., Comment: If you have any comments, find any errors or have any suggestions (in this new version some errors have been corrected, and a better discussion of the case of several complex variables is given), please let us know

In this paper we give optimal lower bounds for the blow-up rate of the $\dot{H}^{s}\left(\mathbb{T}^3\right)$-norm, $\frac{1}{2}\frac{5}{2}$., Comment: This version includes a previous result for the Navier-Stokes equation and we have added an application to a possible blow-up for Euler equations

We review and give elementary proofs of Liouville type properties of harmonic and subharmonic functions in the plane endowed with a complete Riemannian metric, and prove a gap theorem for the possible growth of harmonic functions when this metric has nonnegative Gaussian curvature., Comment: Comments and corrections are very welcome! Mistakes and typos found in a previous version have been corrected

We show for a non homogeneous boundary value problem for the Ricci flow on the disk that when the initial metric has positive curvature and the boundary is convex then the initial metric is deformed, via the normalized flow and along sequences of times, to a metric of constant curvature and totally geodesic boundary. We also show that when the geodesic curvature of the boundary is nonpositive and the metric is rotationally symmetric, the normalized version of the flow exists for all time., Comment: Submitted version. Corrections and suggestions by referees have been incorporated

In this paper we introduce a probabilistic approach to show the existence of initial data with arbitrarily large $L^2(\mathbb{R}^3)$, $\dot{H}^{1/2}(\mathbb{R}^3)$ and $\mathcal{PM}^2$-norms for which a Generalized Navier-Stokes system generate a global regular solution. More precisely, we show that from a certain family of possible large initial data most of them give raise to global regular solutions to a given Generalized Navier-Stokes system., Comment: 21 pages. A better discussion on the probabilistic setting has been included. Conditions on the initial data, as the divergence free condition and real-valuedness, have been taken care of. A section on how the arguments can be generalized has been added