Related papers: A gap theorem on complete shrinking gradient Ricci…

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…

We prove that a complete solution to the Ricci flow on $M\times [-T, 0)$ which has quadratic curvature decay on some end of $M$ and converges locally smoothly to the end of a cone on that neighborhood as $t\nearrow 0$ must be a gradient…

In this paper we study the behavior of the scalar curvature at infinity on complete noncompact steady gradient Ricci solitons. In dimension four, we assume that the canonical Ricci flow induced by the soliton is a weak $\kappa$-solution and…

Let $(M,g)$ be a smooth Riemannian manifold and $\mathsf{G}$ a compact Lie group acting on $M$ effectively and by isometries. It is well known that a lower bound of the sectional curvature of $(M,g)$ is again a bound for the curvature of…

On a smooth $n$-manifold $M$ with $n \geq 3$, we study pairs $(g,T)$ consisting of a Riemannian metric $g$ and a unit length closed vector field $T$. Motivated by how Ricci solitons generalize Einstein metrics via a distinguished vector…

In this paper, I computed the second variation formula of the generalized Einstein-Hilbert functional and prove that a Bismut-flat, Einstein manifold is linearly stable under some curvature assumption. In the last part of the paper, I prove…

We will show that if a gradient shrinking Ricci soliton has an approximate symmetry on one scale, this symmetry propagates to larger scales. This is an example of the shrinker principle which roughly states that information radiates…

We sharpen a gap theorem of Chan & Lee for nonnegative Ricci curvature manifolds that have positive asymptotic volume ratio and small enough scale-invariant integral curvature (so-called "curvature concentration"), by showing that the…

Gradient almost para-Ricci-like solitons on para-Sasaki-like Riemannian $\Pi$-manifolds are studied. It is proved that these objects have constant soliton coefficients. For the soliton under study is shown that the corresponding scalar…

We construct a uniform local bound of curvature operator from local bounds of Ricci curvature and injectivity radius among all $n$-dimensional Ricci flows. Thus new compactness theorems for the Ricci flow and Ricci solitons are derived. In…

We prove that the sharp Li-Yau equality holds for the conjugate heat kernel on shrinking Ricci solitons without any curvature or volume assumptions. This quantity yields several estimates which allows us to classify four dimensional,…

The geometric flow theory and its applications turned into one of the most intensively developing branches of modern geometry. Here, a brief introduction to Finslerian Ricci flow and their self-similar solutions known as Ricci solitons are…

In this paper we consider $\rho$-Einstein solitons of type $M= \left(B^n, g^{*}\right) \times (F^m,g_F)$, where $\left(B^n,g^{*}\right)$ is conformal to a pseudo-Euclidean space and invariant under the action of the pseudo-orthogonal group,…

Let $(Y, d)$ be a Gromov-Hausdorff limit of $n$-dimensional closed shrinking K\"ahler-Ricci solitons with uniformly bounded volumes and Futaki invariants. We prove that off a closed subset of codimension at least 4, Y is a smooth manifold…

By Perelman's $\mathcal L$-geodesic theory, we study the blow-down solutions on a noncompact $\kappa$-noncollapsed steady gradient Ricci soliton $(M^n, g)$ $(n\ge 4)$ with nonnegative curvature operator and positive Ricci curvature away…

In this paper we give some results on the topology of manifolds with $\infty$-Bakry-\'Emery Ricci tensor bounded below, and in particular of steady and expanding gradient Ricci solitons. To this aim we clarify and further develop the theory…

In this paper, we study complete oriented $f$-minimal hypersurfaces properly immersed in a cylinder shrinking soliton $(\mathbb{S}^n\times \mathbb{R}, \bar{g}, f)$. We prove that such hypersurface with $L_f$-index one must be either…

The classical Bach tensor in four dimensions can be expressed as a linear combination of two independent, symmetric, divergence-free, quadratic-in-curvature tensors U and V. Several classification results for gradient-shrinking Ricci…

In this paper, we study the volume growth property of a non-compact complete Riemannian manifold $X$. We improve the volume growth theorem of Calabi (1975) and Yau (1976), Cheeger, Gromov and Taylor (1982). Then we use our new result to…

In this short note, we will prove a volume stability theorem which says that if an n-dimensional toric manifold $M$ admits a $\mathbb{T}^n$ invariant K\"ahler metric $\omega$ with Ricci curvature no less than 1 and its volume is close to…