DOI:10.1080/00029890.2009.11920920 - Corpus ID: 6068179
Manifolds with Density and Perelman's Proof of the Poincaré Conjecture
@article{Morgan2009ManifoldsWD, title={Manifolds with Density and Perelman's Proof of the Poincar{\'e} Conjecture}, author={Frank Morgan}, journal={The American Mathematical Monthly}, year={2009}, volume={116}, pages={134 - 142}, url={https://api.semanticscholar.org/CorpusID:6068179} }
- F. Morgan
- Published in The American mathematical… 1 February 2009
- Mathematics
1. DEFINITIONS. A manifold with density is a Riemannian manifold Mn (n dimensional surface with an infinitesimal arclength ds that you can use to compute lengths, areas, and volumes) with a positive…
41 Citations
41 Citations
Constructions of Helicoidal Surfaces in Euclidean Space with Density
- D. YoonDong-Soo KimYoung Ho KimJae Won Lee
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- 2017
This work considers the Euclidean 3-space R 3 with a positive density function e ϕ and construct all the helicoidal surfaces in the space by solving the second-order non-linear ordinary differential equation with the weighted Gaussian curvature and the mean curvature functions.
Constructions of Helicoidal Surfaces in a 3-Dimensional Complete Manifold with Density
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In this paper, we construct a helicoidal surface with a prescribed weighted mean curvature and weighted extrinsic curvature in a 3-dimensional complete manifold with a positive density function. We…
Helicoidal surfaces in Euclidean space with density
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Our principal goal is to study the prescribed curvature problem in a manifold with density. In particular, we consider a Euclidean 3-space R with a positive density function e, where φ = −x−y, (x, y,…
Some curvature pinching results for Riemannian manifolds with density
- W. Wylie
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- 2015
In this note we consider versions of both Ricci and sectional curvature pinching for Riemannian manifold with density. In the Ricci curvature case the main result implies a diameter estimate that is…
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- B. Klatt
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We extend the well-known formula for the Euler class of a real oriented even-dimensional vector bundle in terms of the curvature of a metric connection to the case of a general linear connection…
Positive weighted sectional curvature
- Lee KennardW. Wylie
- Mathematics
- 2014
In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We…
Weighted minimal translation surfaces in Minkowski 3-space with density
- D. Yoon
- Mathematics, Physics
- 2017
The aim of this work is to study translation surfaces in a Minkowski 3-space ℝ13 with density. Translation surfaces in ℝ13 are defined as the two generating curves which lie in orthogonal planes.…
Type I+ Helicoidal Surfaces with Prescribed Weighted Mean or Gaussian Curvature in Minkowski Space with Density
- Ö. G. YıldızS. HizalM. Akyiğit
- MathematicsAnalele Universitatii "Ovidius" Constanta - Seria…
- 2018
Abstract In this paper, we construct a helicoidal surface of type I+ with prescribed weighted mean curvature and Gaussian curvature in the Minkowski 3−space ℝ13 ${\Bbb R}_1^3$ with a positive density…
Helicoidal Surfaces Which Have the Timelike Axis in Minkowski Space with Density
- Ö. G. YıldızM. ErgütM. Akyiğit
- Mathematics, Physics
- 2018
In this paper, we study the prescribed curvature problem in manifold with density. We consider the Minkowski 3-space with a positive density function. For a given plane curve and an axis in the plane…
Constructions of Type III+ Helicoidal Surfaces in Minkowski Space with Desity
- Ö. G. YıldızM. Akyiğit
- Mathematics, Physics
- 2019
In this paper, we construct a helicoidal surface of type III+ with prescribed weighted mean curvature and weighted Gaussian curvature in the Minkowski 3-space R^3_ 1 with a positive density function.…
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We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric…
Lévy–Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator
Abstract. We establish, by simple semigroup arguments, a Lévy–Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with…
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Below we shall establish certain extremal properties of half-spaces for spherically symmetrical and, in particular, Gaussian (including infinite-dimensional) measures: we also prove inequalities for…
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We consider the free boundary isoperimetric problem in sectors of the Gauss plane. The solution is not always a circular arc as in sectors of the Euclidean plane. We prove that the solution is…
THE ISOPERIMETRIC PROBLEM ON PLANES WITH DENSITY
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We discuss the isoperimetric problem in planes with density. In particular, we examine planes with generalized curvature zero. We solve the isoperimetric problem on the plane with density e x , as…
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be found in many textbooks. I am not that happy with Meyer’s treatment of integration with respect to vector-valued continuous semimartingales, however. This is introduced only as sum of…
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