arXiv subjects

Renan Assimos

Publications and source records attributed to Renan Assimos.

8 recordsLinked to original sources

arXiv2024

Lipschitz regularity of sub-elliptic harmonic maps into CAT(0) spaces

We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the $n$-dimensional Heisenberg group into $CAT(0)$ spaces. Our main theorem establishes that these maps have the desired Lipschitz regularity, extending the Hölder regularity in this setting proven by Y. Gui et. al and obtaining same regularity as in the work of H-C. Zhang and X-P. Zhu for certain sub-Riemannian geometries, see also the works of A. Mondino and D. Semola, as well as N. Gigli, for generalisations to RCD spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.

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arXiv2024

Remarks on the generalised Calabi-Yau problem in higher codimension

By introducing a more flexible notion of convexity, we obtain a new Omori-Yau maximum principle for harmonic maps. In the spirit of the Calabi-Yau conjectures, this principle is more suitable for studying the unboundedness of certain totally geodesic projections of minimal submanifolds of higher codimension. We further explore this maximum principle by applying it to conformal maps, harmonic maps into Cartan-Hadamard manifolds, as well as cone, wedge and halfspace theorems.

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arXiv2023

Perturbed cone theorems for proper harmonic maps

Inspired by the halfspace theorem for minimal surfaces in $\mathbb{R}^3$ of Hoffman-Meeks, the halfspace theorem of Rodriguez-Rosenberg, and the cone theorem of Omori, we derive new non-existence results for proper harmonic maps into perturbed cones in $\mathbb{R}^n$, horospheres in $\mathbb{H}^n$ and also into perturbed Riemannian cones. The technical tool in use is an extension of the foliated maximum principle appearing in Assimos-Jost to the non-compact setting.

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arXiv2022

Graphical mean curvature flow with bounded bi-Ricci curvature

We consider the graphical mean curvature flow of strictly area decreasing maps $f:M\to N$, where $M$ is a compact Riemannian manifold of dimension $m>1$ and $N$ a complete Riemannian surface of bounded geometry. We prove long-time existence of the flow and that the strictly area decreasing property is preserved, when the bi-Ricci curvature $BRic_M$ of $M$ is bounded from below by the sectional curvature $σ_N$ of $N$. In addition, we obtain smooth convergence to a minimal map if $Ric_M\ge\sup\{0,{\sup}_Nσ_N\}$. These results significantly improve known results on the graphical mean curvature flow in codimension $2$.

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arXiv2020

On the intersection of minimal hypersurfaces of $S^k$

It is known since the work of Frankel that two compactly immersed minimal hypersurfaces in a manifold with positive Ricci curvature must have an intersection point. Several generalizations of this result can be found in the literature, for example in the works of Lawson, Petersen and Wilhelm, among others. In the special case of minimal hypersurfaces of $S^k$, we prove a stronger version of Frankel's theorem. Namely, we show that if two compact minimal hypersurfaces $M_1$, $M_2$ of $S^k$ and a point $\mathbf{p}\in S^k$ are given, then $M_1$ and $M_2$ have an intersection point in the hemisphere with respect to $\mathbf{p}$. As a corollary of this result, we give an alternative proof to Ros' two-piece property of minimal surfaces of $S^3$, for the general dimension case.

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arXiv2020

Spherical Bernstein theorems for codimension 1 and 2

A result of B.Solomon (On the Gauss map of an area-minimizing hypersurface. 1984. Journal of Differential Geometry, 19(1), 221-232.) says that a compact minimal hypersurface $M^k$ of the sphere $S^{k+1}$ with $H^1(M)=0$, whose Gauss map omits a neighborhood of an $S^{k-1}$ equator, is totally geodesic in $S^{k+1}$. We develop a new proof strategy which can also obtain an analogous result for codimension 2 compact minimal submanifolds of $S^{k+1}$.

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arXiv2019

Harmonic maps from surfaces of arbitrary genus into spheres

We relate the existence problem of harmonic maps into $S^2$ to the convex geometry of $S^2$. On one hand, this allows us to construct new examples of harmonic maps of degree 0 from compact surfaces of arbitrary genus into $S^2$. On the other hand, we produce new example of regions that do not contain closed geodesics (that is, harmonic maps from $S^1$) but do contain images of harmonic maps from other domains. These regions can therefore not support a strictly convex function. Our construction builds upon an example of W. Kendall, and uses M. Struwe's heat flow approach for the existence of harmonic maps from surfaces.

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arXiv2019

The Geometry of Maximum Principles and a Bernstein Theorem in Codimension 2

Moser's Bernstein theorem \cite{moser61} says that an entire minimal graph of codimension 1 with bounded slope must be a hyperplane. An analogous result for arbitrary codimension is not true, by an example of Lawson-Osserman. Here, we show that Moser's theorem nevertheless extends to codimension 2, i.e., a minimal $p$-submanifold in $R^{p+2}$, which is the graph of a smooth function defined on the entire $R^p$ with bounded slope, must be a $p$-plane. Our method depends on convexity properties of Grassmannians which come into play as the targets of the (harmonic) Gauss maps of our minimal submanifolds. In fact, we develop a general method to construct subsets of complete Riemannian manifolds that cannot contain images of non-constant harmonic maps from compact manifolds. When applied to Grassmannians, for codimension 2, it yields an appropriate domain that cannot contain nontrivial images of such harmonic maps. Our conclusion will then be reached by an application of Allard's theorem to handle the issue that the minimal submanifolds in question are not compact.

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