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In this paper we describe moving frames and differential invariants for curves in two different $| 1 |$ -graded parabolic manifolds $G / H$ , $G = O (p + 1, q + 1)$ and $G = O (2 m, 2 m)$ , and we define differential invariants of projective-type. We then show that, in the first case, there are geometric flows in $G / H$ inducing equations of KdV-type in the projective-type differential invariants when proper initial conditions are chosen. We also show that geometric Poisson brackets in the space of differential invariants of curves in $G / H$ can be reduced...

Proof of the double bubble conjecture.

Hutchings, Michael, Morgan, Frank, Ritoré, Manuel, Ros, Antonio (2000)

Electronic Research Announcements of the American Mathematical Society [electronic only]

Quadratic tilt-excess decay and strong maximum principle for varifolds

Reiner Schätzle (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper, we prove that integral $n$ -varifolds $μ$ in codimension 1 with $H_{μ} \in L_{loc}^{p} (μ)$ , $p > n$ , $p \geq 2$ have quadratic tilt-excess decay ${tiltex}_{μ} (x, ϱ, T_{x} μ) = O_{x} (ϱ^{2})$ for $μ$ -almost all $x$ , and a strong maximum principle which states that these varifolds cannot be touched by smooth manifolds whose mean curvature is given by the weak mean curvature $H_{μ}$ , unless the smooth manifold is locally contained in the support of $μ$ .

Quelques notions simples en géométrie riemannienne et leurs applications aux feuilletages compacts.

Hansklaus Rummler (1979)

Commentarii mathematici Helvetici

Ramification of the Gauss map of complete minimal surfaces in $ℝ^{3}$ and $ℝ^{4}$ on annular ends

Gerd Dethloff, Pham Hoang Ha (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

In this article, we study the ramification of the Gauss map of complete minimal surfaces in $ℝ^{3}$ and $ℝ^{4}$ on annular ends. We obtain results which are similar to the ones obtained by Fujimoto ([4], [5]) and Ru ([13], [14]) for (the whole) complete minimal surfaces, thus we show that the restriction of the Gauss map to an annular end of such a complete minimal surface cannot have more branching (and in particular not avoid more values) than on the whole complete minimal surface. We thus give an improvement...

Ramification of the Gauss map of complete minimal surfaces in $ℝ^{m}$ on annular ends

Gerd Dethloff, Pham Hoang Ha, Pham Duc Thoan (2016)

Colloquium Mathematicae

We study the ramification of the Gauss map of complete minimal surfaces in $ℝ^{m}$ on annular ends. This is a continuation of previous work of Dethloff-Ha (2014), which we extend here to targets of higher dimension.