arXiv research

Noncompact Ricci-flat solutions have infinite unstable dimensions.

The paper constructs a new Ricci-flat metric on almost abelian Lie groups.

Investigates stability of piecewise flat Ricci flow using analysis and simulations.

The paper constructs flat metrics on orbifolds and resolutions.

We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by-product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case, provided the soliton has nonnegative curvature.

New exact spherically symmetric vacuum solutions found in Finsler gravity.

Solves geodesic equations on specific metrics types.

Study gap phenomenon in flat manifolds with Ricci curvature.

Study shows uniqueness of solutions on complex manifolds without requiring solution decay.

Using a recently developed piecewise flat method, numerical evolutions of the Ricci flow are computed for a number of manifolds, using a number of different mesh types, and shown to converge to the expected smooth behaviour as the mesh resolution is increased. The manifolds were chosen to have varying degrees of homoge…

New gravitational solitons and infinite topological manifolds found.

B List has recently studied a geometric flow whose fixed points correspond to static Ricci flat spacetimes. It is now known that this flow is in fact Ricci flow modulo pullback by a certain diffeomorphism. We use this observation to associate to each static Ricci flat spacetime a local Ricci soliton in one higher dimen…

Study of $t$-Gauduchon Ricci-flat condition under Chern-Ricci flow on non-Kähler manifolds.

We give a natural way to identify between two scales, potentially arbitrarily far apart, in a non-compact Ricci-flat manifold with Euclidean volume growth when a tangent cone at infinity has smooth cross section. The identification map is given as the gradient flow of a solution to an elliptic equation.

Study on type-D Ricci-flat metrics with Killing spinors and Killing vectors.

We show that for any solvable Lie group of real type, any homogeneous Ricci flow solution converges in Cheeger-Gromov topology to a unique non-flat solvsoliton, which is independent of the initial left-invariant metric. As an application, we obtain results on the isometry groups of non-flat solvsoliton metrics and Eins…

Study of pseudo-Riemannian metrics related to Monge-Ampère structures.

Local equivalence shown between specific distributions and flat Cartan distribution.

D3-brane solutions derived from Ricci-flat metrics on Kähler-Einstein surfaces.

In this paper, we study the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension one, by using the bracket flow. We prove that solutions to the Ricci flow are immortal, the omega-limit of bracket flow solutions is a single point, and that for any sequence of times there exists a subsequence…

In this article we study the Kähler Ricci flow, the corresponding parabolic Monge Ampère equation and complete non-compact Kähler Ricci flat manifolds. In our main result Theorem \ref{mainthm} we prove that if $(M, g)$ is sufficiently close to being Kähler Ricci flat in a suitable sense, then the Kähler Ricci flow \eqr…

Constructs Ricci-flat K3 metrics using D-geometry.

Conditions for a soliton's dual form to be harmonic or Ricci harmonic are derived.

In this paper, we prove that any non-flat ancient solution to Kähler-Ricci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero. We also prove that any gradient shrinking solitons with positive bisectional curvature must be compact. Both results generalize the corresponding earlier resul…

We study warped products semi-Riemannian Einstein manifolds. We consider the case in that the base is conformal to an n-dimensional pseudo Euclidean space and invariant under the action of an translation group. We provide all such solutions in the case Ricci flat when the base is conformal to an n-dimensional pseudo-Eu…

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of type I $κ$-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman's pr…

Study shows Kähler-Einstein metric singularities linked to curvature.

Ancient solutions found for a specific flow on symplectic half-flat structures.

Study ancient Ricci flows with nonnegative Ricci curvature and their asymptotic geometry.

The paper studies parallel spinor flows on 3D Cauchy hypersurfaces and provides initial data characterizations.

Einstein-Kropina metrics extend Einstein condition to all signatures and classify Finsler gravity solutions.

We construct gradient Kähler-Ricci solitons on Ricci-flat Kähler cone manifolds and on line bundles over toric Fano manifolds. Certain shrinking and expanding solitons are pasted together to form eternal solutions of the Ricci flow. The method we employ is the Calabi ansatz over Sasaki-Einstein manifolds, and the resul…

Existence proved for specific types of gravitational instantons.

The paper explores strong G2-structures with torsion and their geometric properties.

In this paper we show that all conformal metrics to a pseudo-euclidean space invariant under the translation group, and all the conformal metrics product manifold also invariant by translation where F m it is Ricci flat semi-Riemannian manifold, are gradient Ricci almost soliton. We also proved that all conformal metri…

We prove a uniform diameter bound for long time solutions of the normalized Kahler-Ricci flow on an $n$-dimensional projective manifold $X$ with semi-ample canonical bundle under the assumption that the Ricci curvature is uniformly bounded for all time in a fixed domain containing a fibre of $X$ over its canonical mode…

New metrics found on non-Kähler Calabi-Yau manifolds.

Let (M,g) be a three-dimensional steady gradient Ricci soliton which is non-flat and κ-noncollapsed. We prove that (M,g) is isometric to the Bryant soliton up to scaling. This solves a problem mentioned in Perelman's first paper.

New homogeneous manifolds with invariant Bismut Ricci flat connections are constructed.

We show that (a) any entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ${\mathbb C}^{m}$ with the Euclidean metric is flat; (b) any space-like entire graphic self-shrinking solution to the Lagrangian mean curvature flow in ${\mathbb C}^{m}$ with the pseudo-Euclidean metric is flat if the H…

In this paper, we show that any ancient solution to the Ricci flow with the reduced volume whose asymptotic limit is sufficiently close to that of the Gaussian soliton is isometric to the Euclidean space for all time. This is a generalization of Anderson's result for Ricci-flat manifolds. As a corollary, a gap theorem …

We show that the solutions to the second-order differential equation associated to the generalised Chazy equation with parameters $k=2$ and $k=3$ naturally show up in the conformal rescaling that takes a representative metric in Nurowski's conformal class associated to a maximally symmetric $(2,3,5)$-distribution (desc…

Study D3-brane solutions on resolved C^3/Γ singularities, proving metric conjecture.

New graphs with maximum degree 4 found to be Ricci-flat.

In this paper, we study the Ricci flow on closed manifolds equipped with warped product metric $(N\times F,g_{N}+f^2 g_{F})$ with $(F,g_{F})$ Ricci flat. Using the framework of monotone formulas, we derive several estimates for the adapted heat conjugate fundamental solution which include an analog of G. Perelman's dif…

Higher-dimensional Ricci flows are shown to have unique and stable solutions.

Study stability and instability of Ricci-flat metrics under generalized Ricci flow.

Estimates on Einstein manifolds improve Brownian motion behavior and curvature limits.