1. Schwarz-Pick lemma for harmonic maps which are conformal at a pointFranc Forstnerič, David Kalaj, 2024, original scientific article Abstract: We obtain a sharp estimate on the norm of the differential of a harmonic map from the unit disc ${\mathbb D}$ in ${\mathbb C}$ into the unit ball ${\mathbb B}^n$ in ${\mathbb R}^n$, $n\ge 2$, at any point where the map is conformal. In dimension $n=2$, this generalizes the classical Schwarz-Pick lemma, and for $n\ge 3$ it gives the optimal Schwarz-Pick lemma for conformal minimal discs ${\mathbb D}\to {\mathbb B}^n$. This implies that conformal harmonic immersions $M \to {\mathbb B}^n$ from any hyperbolic conformal surface are distance-decreasing in the Poincaré metric on $M$ and the Cayley-Klein metric on the ball ${\mathbb B}^n$, and the extremal maps are precisely the conformal embeddings of the disc ${\mathbb D}$ onto affine discs in ${\mathbb B}^n$. Motivated by these results, we introduce an intrinsic pseudometric on any Riemannian manifold of dimension at least three by using conformal minimal discs, and we lay foundations of the corresponding hyperbolicity theory.
Keywords: harmonic maps, minimal surfaces, Schwarz–Pick lemma, Cayley–Klein metric
Published in DiRROS: 25.04.2024; Views: 59; Downloads: 19
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2. Domains without parabolic minimal submanifolds and weakly hyperbolic domainsFranc Forstnerič, 2023, original scientific article Abstract: We show that if $\Omega$ is an $m$-convex domain in $\mathbb{R}^n$ for some $2 \le m < n$ whose boundary $b\Omega$ has a tubular neighbourhood of positive radius and is not $m$-flat near infinity, then $\Omega$ does not contain any immersed parabolic minimal submanifolds of dimension $\ge m$. In particular, if $M$ is a properly embedded non-flat minimal hypersurface in $\mathbb{R}^n$ with a tubular neighbourhood of positive radius, then every immersed parabolic hypersurface in $\mathbb{R}^n$ intersects $M$. In dimension $n=3$, this holds if $M$ has bounded Gaussian curvature function. We also introduce the class of weakly hyperbolic domains $\Omega$ in $\mathbb{R}^n$, characterised by the property that every conformal harmonic map $\mathbb{C} \to \Omega$ is constant, and we elucidate their relationship with hyperbolic domains, and domains without parabolic minimal surfaces.
Keywords: minimal surfaces, m-plurisubharmonic functions, hyperbolic domain
Published in DiRROS: 10.04.2024; Views: 77; Downloads: 36
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3. The Calabi-Yau problem for minimal surfaces with Cantor endsFranc Forstnerič, 2023, original scientific article Abstract: We show that every connected compact or bordered Riemann surface contains a Cantor set whose complement admits a complete conformal minimal immersion in ${\mathbb R}^3$ with bounded image. The analogous result holds for holomorphic immersions into any complex manifold of dimension at least $2$, for holomorphic null immersions into ${\mathbb C}^n$ with $n \ge 3$, for holomorphic Legendrian immersions into an arbitrary complex contact manifold, and for superminimal immersions into any selfdual or anti-self-dual Einstein four-manifold.
Keywords: minimal surfaces, Calabi–Yau problem, null curve, Legendrian curve
Published in DiRROS: 08.04.2024; Views: 74; Downloads: 35
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4. Proper holomorphic maps in Euclidean spaces avoiding unbounded convex setsBarbara Drinovec-Drnovšek, Franc Forstnerič, 2023, original scientific article Abstract: We show that if $E$ is a closed convex set in $\mathbb C^n$, $n>1$ contained in a closed halfspace $H$ such that ▫$E\cap bH$▫ is nonempty and bounded, then the concave domain $\Omega=\mathbb C^n\setminus E$ contains images of proper holomorphicmaps $f : X \to \mathbb C^n$ from any Stein manifold $X$ of dimension $< n$, with approximation of a givenmap on closed compact subsets of $X$. If in addition $2 {\rm dim} X+1 \le n$ then $f$ can be chosen an embedding, and if $2 {\rm dim} X = n$, then it can be chosen an immersion. Under a stronger condition on $E$, we also obtain the interpolation property for such maps on closed complex subvarieties.
Keywords: Stein manifolds, holomorphic embeddings, Oka manifold, minimal surfaces, convexity
Published in DiRROS: 15.03.2024; Views: 96; Downloads: 48
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5. Minimal surfaces with symmetriesFranc Forstnerič, 2024, original scientific article Abstract: Let $G$ be a finite group acting on a connected open Riemann surface $X$ by holomorphic automorphisms and acting on a Euclidean space ${\mathbb R}^n$ $(n\ge 3)$ by orthogonal transformations. We identify a necessary and sufficient condition for the existence of a $G$-equivariant conformal minimal immersion $F:X\to{\mathbb R}^n$. We show in particular that such a map $F$ always exists if $G$ acts without fixed points on $X$. Furthermore, every finite group $G$ arises in this way for some open Riemann surface $X$ and $n=2|G|$. We obtain an analogous result for minimal surfaces having complete ends with finite total Gaussian curvature, and for discrete infinite groups acting on $X$ properly discontinuously and acting on ${\mathbb R}^n$ by rigid transformations.
Keywords: Riemann surfaces, minimal surfaces, G-equivariant conformal minimal immersion
Published in DiRROS: 13.03.2024; Views: 114; Downloads: 42
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