Noncompact Shrinking 4-Solitons with Nonnegative Curvature

By A. Naber. 2007

75 citations recorded by the source corpus.

We prove the following: Let (M,g,X) be a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M,g) is isometric to R^4 or a finite quotient of S^2xR^2 or S^3xR. In the process we also show that a complete shrinking soliton (M,g,X) with bounded curvature is gradient and k-noncollapsed and the dilation of a Type I singularity is a shrinking soliton. Further in dimension three we show shrinking solitons with bounded curvature can be classified under only the assumption of Rc>= 0.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/3896be0e20c3ecb06668a0c830f0ef1f8299f897

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In Applied Mathematics

1. Ovidiu Munteanu
2. Michele Rimoldi
3. Jiaping Wang
4. Huaihuo Cao
5. A. G. Setti
6. S. Pigola
7. M. Rigoli
8. N. Šešum

In Astronomy and Astrophysics

1. Ovidiu Munteanu
2. Huaihuo Cao
3. Michele Rimoldi
4. Jiaping Wang
5. N. Šešum
6. A. G. Setti
7. S. Pigola
8. M. Rigoli

In Molecular Biology

1. S. Pigola
2. N. Šešum
3. Huaihuo Cao
4. W. Wylie
5. Jiaping Wang
6. Michele Rimoldi
7. Ovidiu Munteanu
8. Qi S. Zhang

Citations in the corpus

• Remarks on non-compact gradient Ricci solitons (2009, background): “The standard Euclidean space (Rm, 〈, 〉 ,∇f) with f (x) = 1 2 A |x|2 + 〈x,B〉 + C, for arbitrary A ∈ R, B ∈ Rm and C ∈ R. Note that f is the essentially unique solution of the equation Hess(f) = A 〈, 〉 on Rm.”
• Ricci almost solitons (2010, result): “We shall prove the next theorem, which extends results obtained in [4, 5, 17, 32].”
• RIGIDITY OF GRADIENT RICCI SOLITONS (2007, unknown): “…r + soliton Cao Chow et al. 2006; Derdzinski 2008] for background on Ricci solitons and their connection to the Ricci flow; we remark that on a compact manifold Ricci solitons are always gradient solitons [Perelman 2002] and that every noncompact shrinking soliton is a gradient soliton [Naber 2007].”
• Geometry of Complete Gradient Shrinking Ricci Solitons (2009, result): “with nonnegative curvature operator and positive isotropic curvature, satisfying certain additional assumptions, is a quotient of either S4 or S3 ×R. Based in part on this result of Ni-Wallach, Naber [31] proved Theorem 2.4. (Naber [31]) A 4-dimensional non-flat complete noncompact shrinking Ricci so”
• Geometry of shrinking Ricci solitons (2014, methodology): “[23] A. Naber, Noncompact Shrinking 4-Solitons with Nonnegative Curvature, J. Reine Angew.”
• On Gradient Ricci Solitons (2009, background): “ity. Since on the other hand, ∆f |∇u| 2 = 2|∇u|∆ f |∇u|+2|∇|∇u|| 2 , it is clear that we get (3.2). Since in particular, ∆f|∇u| ≥ 0 and R M |∇u|2 e−f < ∞, from a standard argument of Yau, see also [22] and Theorem 4.2 in [28] for the f−Laplacian case, it follows that |∇u| = C on M. Then (3.2) imp”
• Recent Progress on Ricci Solitons (2009, background): “s a constant κ = κ(k 0,i 0,T) > 0 such that the solution is κ-noncollapsed on scales ≤ √ T. In the special case of gradient shrinking solitons, the above result has been improved recently by Naber [77] and further by Carrillo-Ni [19]. Proposition 4.1. (Carrillo-Ni [19]) Let (Mn,g,f) be a gradient”
• On the classification of gradient Ricci solitons (2007, methodology): “, and Mantegazza in [12]. We notethat a shrinkingsolitonhas finite fundamental group[32] and that Naber has shown that it can be made into a gradient soliton by adding an appropriate Killing field to X [24]. In the compact case this was proven by Perelman [27]. If we relax the Weyl curvature condition”
• The conjugate heat equation and Ancient solutions of the Ricci flow (2010, background): “Naber [22] proved that M is a gradient shrinking Ricci soliton.”
• On Gradient Ricci Solitons with Symmetry (2007, unknown): “D G ] 2 4 Se p 20 08”