# A geometric problem and the Hopf Lemma. I

• Volume: 008, Issue: 2, page 317-339
• ISSN: 1435-9855

top

## Abstract

top
A classical result of A. D. Alexandrov states that a connected compact smooth $n$-dimensional manifold without boundary, embedded in ${ℝ}^{n+1}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of $M$ in a hyperplane ${X}_{n+1}=\text{const}$ in case $M$ satisfies: for any two points $\left({X}^{\text{'}},{X}_{n+1}\right)$, $\left({X}^{\text{'}},{\stackrel{^}{X}}_{n+1}\right)$ on $M$, with ${X}_{n+1}>{\stackrel{^}{X}}_{n+1}$, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional condition for $n=1$. Some variations of the Hopf Lemma are also presented. Part II [Y. Y. Li and L. Nirenberg, Chinese Ann. Math. Ser. B 27 (2006), 193–218] deals with corresponding higher dimensional problems. Several open problems for higher dimensions are described in this paper as well.

## How to cite

top

Li, Yan Yan, and Nirenberg, Louis. "A geometric problem and the Hopf Lemma. I." Journal of the European Mathematical Society 008.2 (2006): 317-339. <http://eudml.org/doc/277499>.

@article{Li2006,
abstract = {A classical result of A. D. Alexandrov states that a connected compact smooth $n$-dimensional manifold without boundary, embedded in $\mathbb \{R\}^\{n+1\}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of $M$ in a hyperplane $X_\{n+1\}=\text\{const\}$ in case $M$ satisfies: for any two points $(X^\{\prime \},X_\{n+1\})$, $(X^\{\prime \},\hat\{X\}_\{n+1\})$ on $M$, with $X_\{n+1\}>\hat\{X\}_\{n+1\}$, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional condition for $n=1$. Some variations of the Hopf Lemma are also presented. Part II [Y. Y. Li and L. Nirenberg, Chinese Ann. Math. Ser. B 27 (2006), 193–218] deals with corresponding higher dimensional problems. Several open problems for higher dimensions are described in this paper as well.},
author = {Li, Yan Yan, Nirenberg, Louis},
journal = {Journal of the European Mathematical Society},
keywords = {curvature; line-symmetry of a curve; Hopf Lemma; curvature; line-symmetry of a curve; Hopf Lemma},
language = {eng},
number = {2},
pages = {317-339},
publisher = {European Mathematical Society Publishing House},
title = {A geometric problem and the Hopf Lemma. I},
url = {http://eudml.org/doc/277499},
volume = {008},
year = {2006},
}

TY - JOUR
AU - Li, Yan Yan
AU - Nirenberg, Louis
TI - A geometric problem and the Hopf Lemma. I
JO - Journal of the European Mathematical Society
PY - 2006
PB - European Mathematical Society Publishing House
VL - 008
IS - 2
SP - 317
EP - 339
AB - A classical result of A. D. Alexandrov states that a connected compact smooth $n$-dimensional manifold without boundary, embedded in $\mathbb {R}^{n+1}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of $M$ in a hyperplane $X_{n+1}=\text{const}$ in case $M$ satisfies: for any two points $(X^{\prime },X_{n+1})$, $(X^{\prime },\hat{X}_{n+1})$ on $M$, with $X_{n+1}>\hat{X}_{n+1}$, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional condition for $n=1$. Some variations of the Hopf Lemma are also presented. Part II [Y. Y. Li and L. Nirenberg, Chinese Ann. Math. Ser. B 27 (2006), 193–218] deals with corresponding higher dimensional problems. Several open problems for higher dimensions are described in this paper as well.
LA - eng
KW - curvature; line-symmetry of a curve; Hopf Lemma; curvature; line-symmetry of a curve; Hopf Lemma
UR - http://eudml.org/doc/277499
ER -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.