# A geometric problem and the Hopf Lemma. I

Journal of the European Mathematical Society (2006)

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

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topLi, 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 -

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