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We prove existence results for entire graphical translators of the mean curvature flow (the so-called bowl solitons) on Cartan-Hadamard manifolds. We show that the asymptotic behaviour of entire solitons depends heavily on the curvature of the manifold, and that there exist also bounded solutions if the curvature goes to minus infinity fast enough. Moreover, it is even possible to solve the asymptotic Dirichlet problem under certain conditions., Comment: This replaces the first version. We have deleted the whole Section 3 from the previous version due to a gap in a proof. We are grateful to Dr. Hengyu Zhou for pointing out the gap in the proof of Lemma 3.3 in the previous version

Assuming that there exists a translating soliton $u_\infty$ with speed $C$ in a domain $\Omega$ and with prescribed contact angle on $\partial\Omega$, we prove that a graphical solution to the mean curvature flow with the same prescribed contact angle converges to $u_\infty +Ct$ as $t\to\infty$. We also generalize the recent existence result of Gao, Ma, Wang and Weng to non-Euclidean settings under suitable bounds on convexity of $\Omega$ and Ricci curvature in $\Omega$., Comment: This replaces the previous versions. We have added Remark 1.2, Theorem 1.3 and its proof in Section 3

The so called Jenkins–Serrin problem is a kind of Dirichlet problem for graphs with prescribed mean curvature that combines, at the same time, continuous boundary data with regions of the boundary where the boundary values explodes either to \(+\infty \) or to \(-\infty .\) We give a survey on the development of Jenkins–Serrin type problems over domains on Riemannian manifolds. The existence of this type of graphs imposes restrictions on the geometry of the boundary of these domains. We also improve some earlier results by proving Theorem 8, and prove the existence of translating Jenkins–Serrin graphs (Theorem 9).

We give a survey on the development of the study of the asymptotic Dirichlet problem for the minimal surface equation on Cartan–Hadamard manifolds. Part of this survey is based on the introductory part of the doctoral dissertation [29] of the author. The paper is organised as follows. First we introduce Cartan–Hadamard manifolds and the concept of asymptotic Dirichlet problem, then discuss about the development of the results and describe the methods used in the proofs. In the end we mention some results about the nonsolvability of the asymptotic Dirichlet problem.

We study the asymptotic Dirichlet problem for Killing graphs with prescribed mean curvature H in warped product manifolds M× ϱR. In the first part of the paper, we prove the existence of Killing graphs with prescribed boundary on geodesic balls under suitable assumptions on H and the mean curvature of the Killing cylinders over geodesic spheres. In the process we obtain a uniform interior gradient estimate improving previous results by Dajczer and de Lira. In the second part we solve the asymptotic Dirichlet problem in a large class of manifolds whose sectional curvatures are allowed to go to 0 or to - ∞ provided that H satisfies certain bounds with respect to the sectional curvatures of M and the norm of the Killing vector field. Finally we obtain non-existence results if the prescribed mean curvature function H grows too fast., SCOPUS: ar.j, info:eu-repo/semantics/published

We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold $M$ with only one end if $M$ has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constant minimal graphic and $p$-harmonic functions on rotationally symmetric Cartan-Hadamard manifolds under optimal assumptions on the sectional curvatures., The authors are grateful to Professor Luciano Mari who pointed out an error in the proof of the previous version of Proposition 3.1 To appear in Proc. Roy. Soc. Edinburgh Sect. A

We prove the existence of horizontal Jenkins-Serrin graphs that are translating solitons of the mean curvature flow in Riemannian product manifolds $M\times\mathbb{R}$. Moreover, we give examples of these graphs in the cases of $\mathbb{R}^3$ and $\mathbb{H}^2\times\mathbb{R}$., To appear in Rev. Mat. Iberoam

We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan-Hadamard manifold $M$ whose radial sectional curvatures outside a compact set satisfy an upper bound $$K(P)\le - \frac{\phi(\phi-1)}{r(x)^2}$$ and a pointwise pinching condition $$|K(P)|\le C_K|K(P')|$$ for some constants $\phi>1$ and $C_K\ge 1$, where $P$ and $P'$ are any 2-dimensional subspaces of $T_xM$ containing the (radial) vector $\nabla r(x)$ and $r(x)=d(o,x)$ is the distance to a fixed point $o\in M$. We solve the asymptotic Dirichlet problem with any continuous boundary data for dimensions $n>4/\phi+1$., Comment: To appear in J. Geom. Anal

We study the asymptotic Dirichlet problem for $\mathcal{A}$-harmonic functions on a Cartan-Hadamard manifold whose radial sectional curvatures outside a compact set satisfy an upper bound $$ K(P)\le - \frac{1+\varepsilon}{r(x)^2 \log r(x)} $$ and a pointwise pinching condition $$ |K(P)|\le C_K |K(P')| $$ for some constants $\varepsilon>0$ and $C_K\ge 1$, where $P$ and $P'$ are any 2-dimensional subspaces of $T_xM$ containing the (radial) vector $\nabla r(x)$ and $r(x)=d(o,x)$ is the distance to a fixed point $o\in M$. We solve the asymptotic Dirichlet problem with any continuous boundary data $f\in C(\partial_{\infty} M)$. The results apply also to the Laplacian and $p$-Laplacian, $1, Comment: arXiv admin note: substantial text overlap with arXiv:1504.05378

We study the asymptotic Dirichlet problem for $f$-minimal graphs in Cartan-Hadamard manifolds $M$. $f$-minimal hypersurfaces are natural generalizations of self-shrinkers which play a crucial role in the study of mean curvature flow. In the first part of this paper, we prove the existence of $f$-minimal graphs with prescribed boundary behavior on a bounded domain $\Omega \subset M$ under suitable assumptions on $f$ and the boundary of $\Omega$. In the second part, we consider the asymptotic Dirichlet problem. Provided that $f$ decays fast enough, we construct solutions to the problem. Our assumption on the decay of $f$ is linked with the sectional curvatures of $M$. In view of a result of Pigola, Rigoli and Setti, our results are almost sharp., Comment: Final version, to appear in Journal d'Analyse Math\'ematique

This dissertation consists of four research articles whose unifying theme is the existence and non-existence of continuous entire non-constant solutions for nonlinear differential operators on Riemannian manifolds. The existence of such solutions depends heavily on the geometry of the manifold and, in the case of complete and simply connected Riemannian manifolds, we prove the existence under assumptions on the sectional curvature. In the first and fourth article we study the existence of minimal graphic functions by solving the asymptotic Dirichlet problem. Here the idea is to compactify the Cartan-Hadamard manifold by adding an asymptotic boundary and equipping the resulting space with the cone topology. Then one can solve the asymptotic Dirichlet problem i.e. prove the existence of entire solutions with prescribed continuous boundary values on the asymptotic boundary. In the fourth article we prove also a non-existence result by showing that asymptotically non-negative sectional curvature implies uniform gradient estimate for minimal graphic functions with at most linear growth. The second article deals with the existence of A-harmonic functions and the third article with the existence of f-minimal graphs. In the case of A-harmonic functions, we improve an earlier result of A. Vähäkangas by relaxing the assumption on the curvature upper bound. Here, again, we solve the asymptotic Dirichlet problem in order to get the existence result. We solve the asymptotic Dirichlet problem also for the f-minimal equation, but differing from the other papers, here we consider also the existence in the case of bounded domains. Bernsteinin lauseen nojalla koko Euklidisessa avaruudessa määritellyt jatkuvat ja rajoitetut minimipintayhtälön ratkaisut ovat vakioita. Vastaava tulos pätee myös harmonisille, tai yleisemmin A-harmonisille, funktioille globaalin Harnackin epäyhtälön perusteella. Jos Euklidinen avaruus korvataan negatiivisesti kaareutuneella Riemannin monistolla, tilanne muuttuu oleellisesti. Tässä väitöskirjassa tarkastellaan, mitä geometrisia oletuksia avaruudelle tarvitaan ratkaisufunktioiden olemassaolon takaamiseksi. Minimipintayhtälön ratkaisujen olemassaoloa tutkitaan ensimmäisessä ja neljännessä artikkelissa, joista jälkimmäisessä todistetaan myös olemassaolon kannalta kriittinen kaarevuuden yläraja. Minimipintayhtälön ratkaisujen lisäksi neljännessä artikkelissa osoitetaan myös p-harmonisten funktioiden olemassaolo optimaalisen kaarevuuden ylärajan tapauksessa. Väitöskirjan toisessa artikkelissa tarkastellaan A-harmonisia funktioita ja todistetaan, että ehto kaarevuuden alarajalle voidaan korvata tietynlaisella pisteittäisellä kaarevuusoletuksella. Kolmannessa artikkelissa puolestaan todistetaan niin sanottujen f-minimipintojen olemassaolo sekä rajoitetuissa alueissa että koko monistolla. Nämä f-minimipinnat yleistävät perinteisiä minimipintoja, sillä funktio f määrää pinnan keskikaarevuuden ja vakiofunktion tapauksessa yhtälö palautuu minimipintayhtälöksi.