The curvature estimate of gradient ρ Einstein soliton

doi:10.1016/j.geomphys.2020.104063

Journal of Geometry and Physics

Volume 162, April 2021, 104063

https://doi.org/10.1016/j.geomphys.2020.104063 Get rights and content

Abstract

In this paper, we estimate the curvature of the $ρ$ Einstein soliton with $0 \leq ρ < \frac{1}{2 (n - 1)}$ . We show that the curvature operator is at most polynomial growth if the Ricci curvature is bounded and the volume of the unit ball has a uniform lower bound. Furthermore, for 4 dimensional $ρ$ Einstein soliton, the curvature operator is bounded if the Ricci curvature is bounded.

Introduction

In this paper, we consider the following $ρ$ Einstein soliton $R i c + \nabla^{2} f = ρ R g + λ g$ where $(M^{n}, g)$ is a Riemannian manifold, $λ, ρ \in R$ and $f$ is a smooth function on $M^{n}$ . The $ρ$ Einstein soliton is introduced by Catino et al. in [4]. The $ρ$ Einstein soliton is a perturbation of the Ricci soliton equation and gives rise to a self-similar solution to the Ricci Bourguignon flow $\frac{\partial}{\partial t} g = - 2 (R i c - ρ R g) .$ The $ρ$ Einstein soliton equation has been studied by many authors. See [3], [4], [5], [6], [7], [8], [9].

In [10], O. Munteanu and M.D. Wang proved that for any gradient shrinking Ricci soliton with bounded Ricci curvature, the curvature operator has at most polynomial growth. The proof in [10] was based on the integral estimate of the curvature operator and Nash–Moser iteration. The main goal of this paper is to generalize the estimate in [10] to the $ρ$ Einstein soliton. There are some differences between the gradient shrinking Ricci soliton and the gradient shrinking $ρ$ Einstein soliton. In the gradient shrinking Ricci soliton, the principle operator of evolution equation for the curvature operator is the Laplacian operator. This property does not hold in the gradient shrinking $ρ$ Einstein soliton. To overcome this difficulty, we estimate the Weyl tensor instead of the curvature operator. Our first theorem is as follows.

Theorem 1.1

Let $(M, g, f)$ be a gradient shrinking $ρ$ Einstein soliton with $0 \leq ρ < \frac{1}{2 (n - 1)}, R \geq 0$ and such that $| R i c | < K$ , for some positive constant $K$ . Suppose that volume of the unit ball satisfies $V o l B_{x} (1) \geq δ > 0$ for any $x \in M$ . Then the Riemannian curvature tensor grows at most polynomial in the distance function $| R m | (x) \leq C {(r (x) + 1)}^{c}$ for some constant $c > 0$ .

For the 4 dimensional gradient shrinking Ricci soliton, O. Munteanu and J.P. Wang in [11] proves that if the scalar curvature is bounded, then the curvature operator is bounded. In this paper, we adapt the method in [11] and obtain the following result:

Theorem 1.2

Let $(M, g, f)$ be a 4 dimensional gradient shrinking $ρ$ Einstein soliton with $0 \leq ρ < \frac{1}{2 (n - 1)}, R \geq 0$ and such that $| R i c | < K$ , for some positive constant $K$ . Then the Riemannian curvature tensor is bounded $| R m | (x) \leq C, x \in M .$

Section snippets

Curvature equations

In this section, we present the curvature equation on the gradient shrinking $ρ$ Einstein solitons. From [5], we have $(1 - n ρ) R + Δ f = n λ,$ and $(1 - 2 (n - 1) ρ) \nabla R = 2 R i c (\nabla f, \cdot) .$

By taking covariant derivative of Eq. (1), we have $R_{i j, k} + f_{i j, k} = ρ R_{, k} g_{i j}$ Switching $j, k$ of the equality (7), we obtain $ρ (R_{, k} g_{i j} - R_{, j} g_{i k}) = f_{i j, k} - f_{i k, j} + R_{i j, k} - R_{i k, j} = R_{j k i h} f_{h} + R_{i j, k} - R_{i k, j} .$

Now we collect the equations satisfied by curvature operator, Ricci curvature and scalar curvature.

Lemma 2.1

$Δ R_{i j k l} - \nabla_{p} R_{i j k l} \nabla_{p} f = ρ {(H e s s R ⊙ g)}_{i j k l} - Q {(R)}_{i j k l} + 2 ρ R R_{i j k l} + 2 λ R_{i j k l}$ $Δ R_{i k} - R$

Volume growth

Firstly, we recall some estimates for the potential on the $ρ$ Einstein soliton.

Lemma 3.1

[5]

Let $(M, g, f)$ be a gradient shrinking $ρ$ -Einstein soliton with $ρ > 0$ , $R \geq 0$ and such that $| R | < K$ , for some positive constant $K$ . Then, either $f$ is constant or there exist constants $a^{\pm}, b^{\pm}, c^{\pm}$ and $d^{\pm}$ , such that $c^{+} f (r) - d^{+} \leq {| \nabla f |}^{2} (r) \leq a^{+} f (r) + b^{+}, r \geq 0,$ $c^{-} f (r) - d^{-} \leq {| \nabla f |}^{2} (r) \leq a^{-} f (r) + b^{-}, r \leq 0 .$ where $r$ is the signed distance to a connected component $Σ_{0} \subset M$ of some regular level set of $f$ . The constants which appear in the estimates are possibly

Curvature estimates on the $n$ dimensional $ρ$ Einstein solitons

Lemma 4.1

Let $(M^{n}, g, f)$ be a gradient shrinking $ρ$ -Einstein soliton with $0 < ρ \leq \frac{1}{n}, R \geq 0$ and such that $| R i c | \leq K$ for some positive constant $K$ . Suppose that $f$ is not a constant on $M$ . For any $p \geq 3$ there exist some positive constants $C$ and $a > 0$ such that $\int_{M} {| W |}^{p} {(f + 1)}^{- a} \leq C .$

Proof

Define $ϕ (x) = \{\begin{matrix} \frac{r^{2} - f}{r^{2}}, & x \in D (r); \\ 0, & x \in M - D (r) \end{matrix}$ where $D (r) = {\sqrt{f} \leq r}$ . By integration by parts, we have $a \int_{M} {| W |}^{p} {| \nabla f |}^{2} {(f + 1)}^{- a - 1} ϕ^{q} = - \int_{M} {| W |}^{p} \nabla f \cdot \nabla {(f + 1)}^{- a} ϕ^{q} = \int_{M} {| W |}^{p} (Δ f) {(f + 1)}^{- a} ϕ^{q} + \int_{M} {| W |}^{p} {(f + 1)}^{- a} \nabla f \cdot \nabla ϕ^{q} + \int_{M} \nabla {| W |}^{p} \cdot \nabla f {(f + 1)}^{- a} ϕ^{q} .$ By Lemma 3.1, we get $a \frac{{| \nabla f |}^{2} (r)}{f + 1} - Δ f \geq a \frac{c_{1} f (r) - c_{2}}{f (r) + 1} + (1 - n ρ) R$

4 dimensional $ρ$ Einstein shrinking soliton

In this section, we use the index $a, b$ to denote the direction tangent to the level set of $f$ , and $4$ to denote the normal direction to the level set of $f$ . Now, we prove Theorem 1.2.

Proof

From [5], on the regular level set of $f$ , $\nabla R$ is proportional to $\nabla f$ . Hence $\frac{\nabla R}{| \nabla R |} = \frac{\nabla f}{| \nabla f |}$ . The second fundamental form of the level set is $\frac{R_{, a, b}}{| \nabla R |} = h_{a b} = \frac{f_{a b}}{| \nabla f |} .$ Hence $| R_{a, b} | \leq \frac{| \nabla R |}{| \nabla f |} | f_{a, b} | = \frac{| \nabla R |}{| \nabla f |} | ρ R g_{a b} + λ g_{a b} - R_{a b} |,$ Since $(1 - 2 (n - 1) ρ) \nabla R = 2 R i c (\nabla f, \cdot)$ , we have $| R_{, a, b} | \leq c .$

From Eq. (11), we have $R_{, n, n} = Δ R - \sum_{a = 1}^{n - 1} R_{, a, a} = c \nabla R \cdot \nabla f + c R^{2} - c | R i$

Acknowledgments

The second author is supported by the National natural science foundation of China (Grant number: 11971358).

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The curvature estimate of gradient ρ Einstein soliton

Abstract

Introduction

Section snippets

Curvature equations

Volume growth

[5]

Curvature estimates on the n dimensional ρ Einstein solitons

4 dimensional ρ Einstein shrinking soliton

Acknowledgments

Nonlinear Anal.

J. Math. Anal. Appl.

On complete gradient shrinking Ricci solitons

J. Differential Geom.

On the geometry of gradient Einstein-type manifolds

Pacific J. Math.

Rigidity of gradient Einstein shrinkers

Commun. Contemp. Math.

The curvature estimate of gradient $ρ$ Einstein soliton

Curvature estimates on the $n$ dimensional $ρ$ Einstein solitons

4 dimensional $ρ$ Einstein shrinking soliton