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For a complete noncompact Riemannian manifold with nonnegative Ricci curvature, we show that bounded biharmonic functions are constant and the space consists of biharmonic functions with polynomial growth of a fixed rate is finite…

Differential Geometry · Mathematics 2025-11-13 Lin Wang , Miaomiao Zhu

In this paper, we consider complete non-catenoidal minimal surfaces of finite total curvature with two ends. A family of such minimal surfaces with least total absolute curvature is given. Moreover, we obtain a uniqueness theorem for this…

Differential Geometry · Mathematics 2016-03-09 Shoichi Fujimori , Toshihiro Shoda

Ends and end cohomology are powerful invariants for the study of noncompact spaces. We present a self-contained exposition of the topological theory of ends and prove novel extensions including the existence of an exhaustion of a proper…

Algebraic Topology · Mathematics 2025-04-17 William G. Bass , Jack S. Calcut

We introduce a simple procedure to integrate differential forms with arbitrary holomorphic poles on Riemann surfaces. It gives rise to an intrinsic regularization of such singular integrals in terms of the underlying conformal geometry.…

Differential Geometry · Mathematics 2023-04-12 Si Li , Jie Zhou

We propose a new notion called \emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of $p$% -harmonic maps as…

Differential Geometry · Mathematics 2011-01-18 Ye-Lin Ou , Tiffany Troutman , Frederick Wilhelm

This paper investigates positive harmonic functions on a domain which contains an infinite cylinder, and whose boundary is contained in the union of parallel hyperplanes. (In the plane its boundary consists of two sets of vertical…

Classical Analysis and ODEs · Mathematics 2010-10-04 Joanna Pres

Harmonic maps are nonlinear extensions of harmonic functions. They are critical points of natural energy functionals between Riemannian manifolds. Such type of problems appear in Physics, Geometry of Finance and the study of regularity and…

Analysis of PDEs · Mathematics 2023-03-27 Wei Wang

We prove that a finite type curve is an $\xi$-asymptotic line (without parabolic points) of a suitable plane field. It is also given an explicit example of a hyperbolic closed finite type $\xi$-asymptotic line. These results obtained here…

Differential Geometry · Mathematics 2019-08-12 Douglas H. da Cruz , Ronaldo A. Garcia

In this paper, we study the space of translational limits T(M) of a surface M properly embedded in R^3 with nonzero constant mean curvature and bounded second fundamental form. There is a natural map T which assigns to any surface M' in…

Differential Geometry · Mathematics 2008-05-13 William H. Meeks , Giuseppe Tinaglia

We prove that a complete noncompact K\"ahler surface with positive and bounded sectional curvature is biholomorphic to $\mathbb{C}^2$. This result confirms a special case of Yau's conjecture that a complete noncompact K\"ahler $n$-manifold…

Differential Geometry · Mathematics 2025-11-11 Ved Datar , Vamsi Pritham Pingali , Harish Seshadri

We study complete minimal surfaces in $\mathbb{R}^n$ with finite total curvature and embedded planar ends. After conformal compactification via inversion, these yield examples of surfaces stationary for the Willmore bending energy…

Differential Geometry · Mathematics 2024-07-02 Jonas Hirsch , Rob Kusner , Elena Mäder-Baumdicker

Generalising an example by Girondo and Wolfart, we use finite group theory to construct Riemann surfaces admitting two or more regular dessins (i.e. orientably regular hypermaps) with automorphism groups of the same order, and in many cases…

Combinatorics · Mathematics 2011-04-06 Gareth A. Jones

We classify all harmonic maps with finite uniton number from a Riemann surface into an arbitrary compact simple Lie group $G$, whether $G$ has trivial centre or not, in terms of certain pieces of the Bruhat decomposition of the group…

Differential Geometry · Mathematics 2014-05-16 Nuno Correia , Rui Pacheco

We detail the theory of Discrete Riemann Surfaces. It takes place on a cellular decomposition of a surface, together with its Poincar\'e dual, equipped with a discrete conformal structure. A lot of theorems of the continuous theory follow…

Complex Variables · Mathematics 2008-02-13 Christian Mercat

We show the existence of 1-parameter families of non-periodic, complete, embedded minimal surfaces in euclidean space with infinitely many parallel planar ends. In particular we are able to produce finite genus examples and quasi-periodic…

Differential Geometry · Mathematics 2010-12-01 Filippo Morabito , Martin Traizet

Let $T$ be a triangulation of a Riemann surface. We show that the 1-skeleton of $T$ may be oriented so that there is a global bound on the outdegree of the vertices. Our application is to construct extremal metrics on triangulations formed…

Geometric Topology · Mathematics 2012-02-23 William E. Wood

For an embedded singly periodic minimal surface M with genus bigger than or equal to 4 and annular ends, some weak symmetry hypotheses imply its congruence with one of the Hoffman-Wohlgemuth examples. We give a very geometrical proof of…

Differential Geometry · Mathematics 2008-06-12 Valerio Ramos-Batista , Plinio Simoes

The main goal of this article is to relate asymptotic geometric properties on a tower of coverings of a non-compact K\"ahler manifold of finite volume with reasonable geometric assumptions to its universal covering. Applicable examples…

Algebraic Geometry · Mathematics 2012-04-27 Sai-Kee Yeung

We characterize the smallest finite spaces with the same homotopy groups of the spheres. Similarly, we describe the minimal finite models of any finite graph. We also develop new combinatorial techniques based on finite spaces to study…

Algebraic Topology · Mathematics 2007-05-23 Jonathan Ariel Barmak , Elias Gabriel Minian

We study finite order invariants of null-homotopic immersions of a closed orientable surface into an aspherical orientable 3-manifold. We give the foundational constructions, and classify all order one invariants.

Geometric Topology · Mathematics 2007-05-23 Tahl Nowik
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