Gradient Steady Ricci Solitons Research Articles

An Optimal Volume Growth Estimate for Noncollapsed Steady Gradient Ricci Solitons

In this paper, we prove a volume growth estimate for steady gradient Ricci solitons with bounded Nash entropy. We show that such a steady gradient Ricci soliton has volume growth rate no smaller than $$r^{\frac{n+1}{2}}.$$ This result not only improves the estimate in (Chan et al., arXiv:2107.01419 , 2021, Theorem 1.3), but also is optimal since the Bryant soliton and Appleton’s solitons (Appleton, arXiv:1708.00161 , 2017) have exactly this growth rate.

On the classification of four-dimensional gradient Ricci solitons

In this paper, we prove some classification results for four-dimensional gradient Ricci solitons. For a four-dimensional gradient shrinking Ricci soliton with div4Rm±=0, we show that it is either Einstein or a finite quotient of R4, S2×R2 or S3×R. The same result can be obtained under the condition of div4W±=0. We also present some classification results of four-dimensional complete non-compact gradient expanding Ricci soliton with non-negative Ricci curvature and gradient steady Ricci solitons under certain curvature conditions.

On a dichotomy of the curvature decay of steady Ricci solitons

We establish a dichotomy on the curvature decay for four dimensional complete noncompact non Ricci flat steady gradient Ricci solitons with linear curvature decay and proper potential function. A similar dichotomy is also shown in higher dimensions under the additional assumption that the Ricci curvature is nonnegative outside a compact subset. As an application, we prove some classification results on steady gradient Ricci solitons with fast curvature decay and obtain a dichotomy on the asymptotic geometry at spatial infinity.

On four-dimensional steady gradient Ricci solitons that dimension reduce

In this paper, we will study the asymptotic geometry of 4-dimensional steady gradient Ricci solitons under the condition that they dimension reduce to 3-manifolds. We will show that such solitons either strongly dimension reduce to a spherical space form S3/Γ or weakly dimension reduce to the 3-dimensional Bryant soliton. We also show that 4-dimensional steady gradient Ricci soliton singularity models with nonnegative Ricci curvature outside a compact set either are Ricci-flat ALE 4-manifolds or dimension reduce to 3-dimensional manifolds. As a further application, we prove that any steady gradient Kähler-Ricci soliton singularity models on complex surfaces with nonnegative Ricci curvature outside a compact set must be hyperkähler ALE 4-manifolds.

Four-dimensional steady gradient Ricci solitons with 3-cylindrical tangent flows at infinity

In this paper we consider 4-dimensional steady soliton singularity models, i.e., complete steady gradient Ricci solitons that arise as the rescaled limit of a finite time singular solution of the Ricci flow on a closed 4-manifold. In particular, we study the geometry at infinity of such Ricci solitons under the assumption that their tangent flow at infinity is the product of R with a 3-dimensional spherical space form. We also classify the tangent flows at infinity of 4-dimensional steady soliton singularity models in general.

On complete gradient steady Ricci solitons with vanishing 𝐷-tensor

In this paper, we extend the work by the first author and Q. Chen [Duke Math. J. 162 (2013), pp. 1149–1169] to classify n n -dimensional ( n ≥ 5 n\ge 5 ) complete D D -flat gradient steady Ricci solitons. More precisely, we prove that any n n -dimensional complete noncompact gradient steady Ricci soliton with vanishing D-tensor is either Ricci-flat, or isometric to the Bryant soliton. Furthermore, the proof extends to the shrinking case and the expanding case as well.

Steady Ricci solitons with horizontally ϵ-pinched Ricci curvature

In this paper, we prove that any κ-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally ϵ-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any κ-noncollapsed gradient steady Ricci soliton (Mn, g, f) with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature R(x) limr(x)→∞R(x)f(x) = C0 supx∈MR(x) with satisfies $$C_{0} > \frac{n-2}{2}$$ .

Classification of gradient steady Ricci solitons with linear curvature decay

In this paper, we give a description for steady Ricci solitons with a linear decay of sectional curvature. In particular, we classify all 3-dimensional steady Ricci solitons and 4-dimensional $\kappa$-noncollpased steady Ricci solitons with nonnegative sectional curvature under the linear curvature decay.

Curvature estimates for steady Ricci solitons

We show that for an n n dimensional complete non-Ricci flat gradient steady Ricci soliton with potential function f f bounded above by a constant and curvature tensor R m Rm satisfying lim ¯ r → ∞ r | R m | > 1 5 \overline {\lim }_{r\to \infty } r|Rm|>\frac {1}{5} , | R m | ≤ C e − r |Rm|\leq Ce^{-r} for some constant C > 0 C>0 , improving on a result of Munteanu, Sung, and Wang’s. For any four dimensional complete non-Ricci flat gradient steady Ricci soliton with scalar curvature S → 0 S\to 0 as r → ∞ r\to \infty , we prove that | R m | ≤ c S |Rm|\leq cS for some constant c > 0 c>0 , improving on an estimate by Cao and Cui. As an application, we show that for a four dimensional complete non-Ricci flat gradient steady Ricci soliton, | R m | |Rm| decays exponentially provided that lim ¯ r → ∞ r S \overline {\lim }_{r\to \infty } rS is sufficiently small and f f is bounded above by a constant.

Generalized Ricci flow II: Existence for complete noncompact manifolds

In this paper, we continue to study the generalized Ricci flow. We give a criterion on steady gradient Ricci soliton on complete and noncompact Riemannian manifolds that is Ricci-flat, and then introduce a natural flow whose stable points are Ricci-flat metrics. Modifying the argument used by Shi and List, we prove the short time existence and higher order derivatives estimates.

Poisson equation on complete manifolds

We develop heat kernel and Green's function estimates for manifolds with positive bottom spectrum. The results are then used to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds with Ricci curvature bounded below. As an application, we show that the curvature of a steady gradient Ricci soliton must decay exponentially if it decays faster than linear and the potential function is bounded above.

Curvature Estimates for Four-Dimensional Gradient Steady Ricci Solitons

In this paper, we derive certain curvature estimates for 4-dimensional gradient steady Ricci solitons either with positive Ricci curvature or with scalar curvature decay.

Classification of warped product pointwise semi-slant submanifolds in complex space forms

The main principle of this paper is to show that, a warped product pointwise semi-slant submanifold of type Mn = Nn1 T xf Nn2? in a complex space form ?M2m (C) admitting shrinking or steady gradient Ricci soliton, whose potential function is a well-define warped function, is an Einstein warped product pointwise semi-slant submanifold under extrinsic restrictions on the second fundamental form inequality attaining the equality in [4]. Moreover, under some geometric assumption, the connected and compactness with nonempty boundary are treated. In this case, we propose a necessary and sufficient condition in terms of Dirichlet energy function which show that a connected, compact warped product pointwise semi-slant submanifold of complex space forms must be a Riemannian product. As more applications, for the first one, we prove that Mn is a trivial compact warped product, when the warping function exist the solution of PDE such as Euler-Lagrange equation. In the second one, by imposing boundary conditions, we derive a necessary and sufficient condition in terms of Ricci curvature, and prove that, a compact warped product pointwise semi-slant submanifold Mn of a complex space form, is either a CR warped product or just a usual Riemannian product manifold. We also discuss some obstructions to these constructions in more details.

Level set flow in 3D steady gradient Ricci solitons

Given a nontrivial 3-dimensional steady gradient Ricci soliton, if the scalar curvature has decay order between $$-b$$ and $$-a$$ for some $$a\in (0,1], b\ge a$$ , then the umbilical ratio of the level set of the potential function lies in the class $$O(r^{6a-\frac{8a^2}{b}})\cap O(r^{2b-4a})$$ .

Ridigity of Ricci solitons with weakly harmonic Weyl tensors

In this paper, we prove rigidity results on gradient shrinking or steady Ricci solitons with weakly harmonic Weyl curvature tensors. Let (Mn,g,f) be a compact gradient shrinking Ricci soliton satisfying Ric g+Ddf=I g with I >0 constant. We show that if (M,g) satisfies I´W(·,·,∇f)=0, then (M,g) is Einstein. Here W denotes the Weyl curvature tensor. In the case of noncompact, if M is complete and satisfies the same condition, then M is rigid in the sense that M is given by a quotient of product of an Einstein manifold with Euclidean space. These are generalizations of the previous known results in and . Finally, we prove that if (Mn,g,f) is a complete noncompact gradient steady Ricci soliton satisfying I´W(·,·,∇f)=0, and if the scalar curvature attains its maximum at some point in the interior of M, then either (M,g) is flat or isometric to a Bryant Ricci soliton. The final result can be considered as a generalization of main result in .

Infinitesimal rigidity of collapsed gradient steady Ricci solitons in dimension three

The only known example of collapsed three-dimensional complete gradient steady Ricci solitons so far is the 3D cigar soliton $N^2\times \mathbb{R}$, the product of Hamilton's cigar soliton $N^2$ and the real line $\mathbb{R}$ with the product metric. R. Hamilton has conjectured that there should exist a family of collapsed positively curved three-dimensional complete gradient steady solitons, with $\mathsf{S}^1$-symmetry, connecting the 3D cigar soliton. In this paper, we make the first initial progress and prove that the infinitesimal deformation at the 3D cigar soliton is non-essential. In Appendix A, we show that the 3D cigar soliton is the unique complete nonflat gradient steady Ricci soliton in dimension three that admits two commuting Killing vector fields.

Some rigidity results for noncompact gradient steady Ricci solitons and Ricci-flat manifolds

Gradient steady Ricci solitons are natural generalizations of Ricci-flat manifolds. In this article, we prove a curvature gap theorem for gradient steady Ricci solitons with nonconstant potential functions; and a curvature gap theorem for Ricci-flat manifolds, removing the volume growth assumptions in known results.

Gradient Ricci Solitons with Structure of Warped Product

In this paper we consider semi-Riemannian warped product gradient Ricci solitons. We prove that the potential function depends only on the base and the fiber is necessarily Einstein manifold. We provide all such solutions in the case of steady gradient Ricci solitons when the base is conformal to an n-dimensional pseudo-Euclidean space, invariant under the action of an (n − 1)-dimensional translation group, and the fiber is Ricci-flat.

Liouville type theorems for the p-harmonic functions on certain manifolds

We show that the Dirichlet problem at infinity is unsolvable for the p-Laplace equation for any nonconstant continuous boundary data, for certain range of p>n, on an n-dimensional Cartan-Hadamard manifold constructed from a complete noncompact shrinking gradient Ricci soliton. Using the steady gradient Ricci soliton, we find an incomplete Riemannian metric on ${\mathbb R}^2$ with positive Gauss curvature such that every positive p-harmonic function must be constant for $p\geq 4$.

Martin compactification of a complete surface with negative curvature

In this paper we consider the Martin compactification, associated with the operator $$\mathcal {L}= \Delta -1$$ , of a complete non-compact surface $$\Sigma ^2$$ with negative curvature. In particular, we investigate positive eigenfunctions with eigenvalue one of the Laplace operator $$\Delta $$ of $$\Sigma ^2$$ , and prove a uniqueness result: such eigenfunctions are unique up to a positive multiplicative constant, if they vanish on the part of the geometric boundary $$S_\infty (\Sigma ^2)$$ of $$\Sigma ^2$$ where the curvature is bounded above by a negative constant, and satisfy some growth estimate on the other part of $$S_\infty (\Sigma ^2)$$ where the curvature approaches zero. This uniqueness result plays an essential role in our recent paper (Cao and He, Infinitesimal rigidity of steady gradient Ricci soliton in dimension three. arXiv:1412.2714 , 2014, preprint), in which we prove an infinitesimal rigidity theorem for deformations of certain three-dimensional collapsed gradient steady Ricci soliton with a non-trivial Killing vector field.