On Gradient Ricci Solitons

By Ovidiu Munteanu, N. Šešum. 2009

209 citations recorded by the source corpus.

In the first part of the paper we derive integral curvature estimates for complete gradient shrinking Ricci solitons. Our results and the recent work in M. Fernandez-Lopez and E. Garcia-Rio, Rigidity of shrinking Ricci solitons in Math. Z. (2011) classify complete gradient shrinking Ricci solitons with harmonic Weyl tensor. In the second part of the paper we address the issue of existence of harmonic functions on gradient shrinking Kähler and gradient steady Ricci solitons. Consequences to the structure of shrinking and steady solitons at infinity are also discussed.

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

View Paper: https://www.semanticscholar.org/paper/0f24bf07a12b332e45109ee03ad2d19f806f8f17

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In Applied Mathematics

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

In Astronomy and Astrophysics

1. Lu Wang
2. Qiang Chen
3. Ovidiu Munteanu
4. Brett Kotschwar
5. Jiaping Wang
6. Huaihuo Cao
7. Chenxu He
8. Binglong Chen

In Geometry and Topology

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

Citations in the corpus

• On Bach-flat gradient shrinking Ricci solitons (2011, methodology): “artially supported by NSF Grant DMS-0909581. 1 2 HUAI-DONG CAO AND QIANG CHEN the works of Eminenti-La Nave-Mantegazza [15], Petersen-Wylie [24], X. Cao, B. Wang and Z. Zhang [10], and Munteanu-Sesum [20]). Moreover, it follows from the works of Ferna´ndez-L´opez and Garc´ıa-R´ıo [16], and Munteanu-”
• Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, background): “Also, from [16], [26], [30], [38], [43], [46], and [53], it follows that the only complete nonflat shrinking solitons of vanishing Weyl tensor (even harmonic Weyl tensor) in dimensions n ≥ 4 are finite quotients of the standard metrics on S or Sn−1 × R.”
• A Compactness Theorem for Complete Ricci Shrinkers (2010, unknown): “The next lemma, first observed by Munteanu-Sesum [27], will be very useful in the following.”
• On locally conformally flat gradient steady Ricci solitons (2009, background): “mediate corollary, we have Lemma 2.2. Let (Mn,gij,F) be a complete gradient steady soliton. Then it has nonnegative scalar curvature R≥ 0. 6 HUAI-DONG CAO AND QIANG CHEN We remark that Munteanu-Sesum [19] recently proved a Liouville type theorem for gradient steady solitons, namely a gradient steady”
• Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature (2021, unknown): “In particular, Fernández-López and García-Río [21] together with Munteanu and Sesum [28] proved that any n -dimensional complete gradient shrinker with harmonic Weyl tensor is rigid, i.e., it is isometric to a (finite) quotient of N × (cid:82) k , the product soliton of an Einstein manifold N of…”
• Geometry of manifolds with densities (2012, background): “ rate a ≥ 0. Then the weighted volume entropy hf(M) of M,g,e−fdv satisfies hf (M) := limsup r→∞ lnVolf (Bp (r)) r ≤ a. The estimate for hf (M) is in fact sharp for any gradient steady Ricci soliton by [21]. Our goal here is to understand the structure of M when this volume entropy is maximal. Recall ”