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In this paper, we consider a connected orientable closed Riemannian manifold $M^{n+1}$ with positive Ricci curvature. Suppose $G$ is a compact Lie group acting by isometries on $M$ with $3\leq {\rm codim}(G\cdot p)\leq 7$ for all $p\in M$.…

Let $F_g$ denote the closed orientable surface of genus $g$. What is the least order finite group, $G_g$, for which there is a homomorphism $\psi$ from $\pi_1(F_g)$ to $G_g$ so that no nontrivial simple closed curve on $F_g$ represents an…

Let $\Gamma_{g,1}^m$ be the mapping class group of the orientable surface $\Sigma_{g,1}^m$ of genus $g$ with one parametrised boundary curve and $m$ permutable punctures; when $m=0$ we omit it from the notation. Let…

Let $\Gamma_g$ denote the orientation-preserving Mapping Class Group of the genus $g\geq 1$ closed orientable surface. In this paper we show that for fixed $g$, every finite group occurs as a quotient of a finite index subgroup of…

The isoperimetric ratio of an embedded surface in $R^3$ is defined as the ratio of the area of the surface to power three to the squared enclosed volume. The aim of the present work is to study the minimization of the Willmore energy under…

It is known that any periodic map of order $n$ on a closed oriented surface of genus $g$ can be equivariantly embedded into $S^m$ for some $m$. In the orientable and smooth category, we determine the smallest possible $m$ when $n\geq 3g$.…

In this paper, we construct a one-parameter family of minimal surfaces in the Euclidean $3$-space of arbitrarily high genus and with three ends. Each member of this family is immersed, complete and with finite total curvature. Another…

K3-surfaces with antisymplectic involution and compatible symplectic actions of finite groups are considered. In this situation actions of large finite groups of symplectic transformations are shown to arise via double covers of Del Pezzo…

Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let $\mathbf{G}$ be a semisimple algebraic $\mathbb{R}$-group…

We discover a family of closed, embedded minimal surfaces in the three-dimensional round sphere which includes new examples with low genus. The existence proof relies on an equivariant min-max procedure applied to a novel sweepout which is…

Let $k$ be a number field and $B$ be a central simple algebra over $k$ of dimension $p^2$ where $p$ is prime. In the case that $p=2$ we assume that $B$ is not totally definite. In this paper we study sets of pairwise nonisomorphic maximal…

We consider a partially overdetermined problem in a sector-like domain $\Omega$ in a cone $\Sigma$ in $\mathbb{R}^N$, $N\geq 2$, and prove a rigidity result of Serrin type by showing that the existence of a solution implies that $\Omega$ is…

Putman and Wieland conjectured that if $\tilde{\Sigma} \rightarrow \Sigma$ is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of $H_1(\tilde{\Sigma};\mathbb{Q})$…

Let $\Sigma_g$ denote the closed orientable surface of genus $g$ and fix an arbitrary simplicial triangulation of $\Sigma_g$. We construct and study a natural surjective group homomorphism from the surface braid group on $n$ strands on…

We show that a 3-manifold containing an incompressible surface has topologically minimal surfaces of arbitrary high genus.

We consider large finite group-actions on surfaces and discuss and compare various notions for such actions: Hurwitz actions and Hurwitz groups; maximal reducible and completely reducible actions; bounding and geometrically bounding…

If K is an odd-dimensional flag closed manifold, flag generalized homology sphere or a more general flag weak pseudomanifold with sufficiently many vertices, then the maximal number of edges in K is achieved by the balanced join of cycles.…

We generalize the following result of White: Suppose $N$ is a compact, strictly convex domain in $\RR^3$ with smooth boundary. Let $\Sigma$ be a compact 2-manifold with boundary. Then a generic smooth curve $\Gamma\cong \partial\Sigma$ in…

We explain how the spectrum of a closed embedded surface $\Sigma \subset \mathbb{R}^3$ relates to the Dirichlet spectrum of the bounded domain $\Omega \subset \mathbb{R}^3$ with $\partial \Omega = \Sigma$. We prove that there exists a…

This paper focuses on the classification of classes of topological equivalence of finite group actions on Riemann surfaces. By the Riemann-Hurwitz bound, there are just finitely many groups that act conformally on a closed orientable…