On a classification of the gradient shrinking solitons

By Lei Ni, Nolan Wallach. 2007

152 citations recorded by the source corpus.

The main purpose of this article is to provide an alternate proof to a result of Perelman on gradient shrinking solitons. In dimension three we also generalize the result by removing the $\kappa$-non-collapsing assumption. In high dimension this new method allows us to prove a classification result on gradient shrinking solitons with vanishing Weyl curvature tensor, which includes the rotationally symmetric ones.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1e811bd3c5edf8965147dbb1e0ce57f551d83884

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Ovidiu Munteanu
2. Huaihuo Cao
3. G. Catino
4. Jiaping Wang
5. Qiang Chen
6. Brett Kotschwar
7. L. Mazzieri
8. Lu Wang

In Astronomy and Astrophysics

1. Ovidiu Munteanu
2. Huaihuo Cao
3. Qiang Chen
4. Jiaping Wang
5. G. Catino
6. Lu Wang
7. Brett Kotschwar
8. Zhuhong Zhang

In Geometry and Topology

1. Ovidiu Munteanu
2. Huaihuo Cao
3. G. Catino
4. Jiaping Wang
5. Qiang Chen
6. Lu Wang
7. L. Mazzieri
8. Brett Kotschwar

Citations in the corpus

• On Gradient Ricci Solitons with Symmetry (2007, unknown): “D G ] 2 4 Se p 20 08”
• On Bach-flat gradient shrinking Ricci solitons (2011, background): “For higher dimensions, it has been proven that complete locally conformally flat gradient shrinking Ricci solitons are finite quotients of either the round sphere S n , or the Gaussian shrinking soliton R n , or the round cylinder S n − 1 × R (first due to Z. H. Zhang [26] based on the work of Ni-Walla”
• On the classification of gradient Ricci solitons (2007, background): “Recently Ni and Wallach [ 26 ] have given an alternative approach to proving the classification of 3-dimensional shrinkers which extends to higher dimensional manifolds with zero Weyl tensor.”
• GRADIENT SHRINKING SOLITONS WITH VANISHING WEYL TENSOR (2008, methodology): “outside of a compact set, where c t [12].”
• On the geometry of gradient einstein-type manifolds (2014, unknown): “[25], [28], [39], [22], [38], [48], [37], [40], [10], [7], [14], [42], [11], [13], [8], [6], [33], [5] and references therein).”
• Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, background): “Hamilton [33] proved that the only complete nonflat twodimensional shrinking solitons are the standard round metrics on S2 and RP 2, and, with a combination of results from his later paper [34] and the work of Ivey [35], Perelman [47], Ni-Wallach [46], and Cao-ChenZhu [16], it follows that the only ”
• Geometry of shrinking Ricci solitons (2014, methodology): “[24] L. Ni and N. Wallach, On a classification of gradient shrinking solitons, Math.”
• Rigidity of gradient Einstein shrinkers (2013, methodology): “In dimension three, a complete classification of gradient Ricci shrinkers was given by Ivey [6] in the compact case and by Perelman [11], Ni-Wallach [10] and Cao-ChenZhu [2] in the complete case.”
• On complete gradient shrinking Ricci solitons (2009, result): “We also point out that it has been known, by the work of Ni-Wallach [12] and Cao-Chen-Zhu [3], that any 3- dimensional complete noncompact non-flat shrinking gradient soliton is necessarily the round cylinder S2 × R or one of its Z2 quotients.”