Sharp Gaussian bounds derived for Schrödinger kernel on Ricci solitons.
problem Analyzing Schrödinger heat kernel on gradient shrinking Ricci solitons.
method Deriving sharp Gaussian upper bounds for the Schrödinger heat kernel.
result Sharp upper and lower bounds for eigenvalues of the Schrödinger operator.
The study finds complete translating solitons for certain powers of Gaussian curvature in Riemannian products.
problem Exploring translating solitons in Riemannian products with powers of Gaussian curvature.
method Investigating Kα-flows in Riemannian products MimesR for M=Rn,Sn,HFm. result Existence of complete rotational translating solitons for certain values of α in MimesR. In this paper, we classify n-dimensional (n>3) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat hence a finite quotient of the Gaussian shrinking soliton R4 or the round cyl…
Estimates heat equation on shrinking Ricci solitons with uniform bounds.
problem Analyzing heat equation on shrinking Ricci solitons.
method Proved L2 estimate with time-dependent Gaussian weight. result Uniform bounds for heat equation along Ricci flow.
Study on non-gradient Ricci almost solitons in warped products.
problem Understanding non-gradient Ricci almost solitons.
method Construction method and explicit example in warped products.
result Rigidity result for Gaussian soliton.
4D gradient solitons with constant curvature are rigid.
problem Characterizing 4D gradient Ricci solitons with constant scalar curvature.
method Proving rigidity using constant scalar curvature and quotient structures.
result 4D gradient solitons with constant curvature are rigid.
The paper studies geometric properties of soliton surfaces using an extended Darboux frame field.
problem Geometric analysis of soliton surfaces associated with the Betchov-Da Rios equation.
method Derivative formulas of an extended Darboux frame field, geometric invariants, curvature calculations.
result Construction of curvature ellipse and Wintgen ideal soliton surfaces.
The second del Pezzo surface is known by work of Tian-Zhu and Wang-Zhu to admit a unique Kaehler-Ricci soliton. Applying a method described in hep-th/0703057, we use Ricci flow to numerically compute that soliton metric. We numerically compute the value of its Perelman entropy (or Gaussian density).
We show that if two gradient Ricci solitons are asymptotic along some end of each to the same regular cone, then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does no…
We first show that a Kähler cone appears as the tangent cone of a complete expanding gradient Kähler-Ricci soliton with quadratic curvature decay with derivatives if and only if it has a smooth canonical model (on which the soliton lives). This allows us to classify two-dimensional complete expanding gradient Kähler-Ri…
The paper examines soliton surfaces using a parallel transport frame field in 4D space.
problem Geometric properties of soliton surfaces associated with the Betchov-Da Rios equation.
method Parallel transport frame field approach in four-dimensional Euclidean space.
result Characterization of soliton surfaces as flat, minimal, semi-umbilic, or Wintgen ideal.
The study examines four-dimensional gradient Ricci solitons and their properties.
problem Characterizing four-dimensional complete gradient shrinking Ricci solitons.
method Proving properties and providing curvature estimates for solitons under specific conditions.
result Conditions for four-dimensional complete gradient shrinking Ricci solitons.
The study classifies and disproves gradient properties of certain solitons on specific Lie groups.
problem Characterizing and proving non-graduation of solitons on specific Lie groups.
method Proving structure theorems and analyzing specific examples of solitons.
result Examples of solitons that cannot be made gradient, including specific Lie groups.
We prove that a gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is rigid. For the 4-dimensional case, we show that any gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is either Einstein, or a finite quotient of the Gaussian shrinking soliton $\…
In this paper we prove new classification results for nonnegatively curved gradient expanding and steady Ricci solitons in dimension three and above, under suitable integral assumptions on the scalar curvature of the underlying Riemannian manifold. In particular we show that the only complete expanding solitons with no…
We classify complete gradient Ricci solitons satisfying a fourth-order vanishing condition on the Weyl tensor, improving previously known results. More precisely, we show that any n-dimensional (n≥4) gradient shrinking Ricci soliton with fourth order divergence-free Weyl tensor is either Einstein, or a finite q…
In previous work, the authors studied the linear stability of algebraic Ricci solitons on simply connected solvable Lie groups (solvsolitons), which are stationary solutions of a certain normalization of Ricci flow. Many examples were shown to be linearly stable, leading to the conjecture that all solvsolitons are line…
The paper examines the rigidity of eigenvalues in shrinking Ricci solitons.
problem Rigidity of eigenvalues in shrinking Ricci solitons.
method Analysis of the drifted Laplacian on shrinking Ricci solitons, showing eigenvalue bounds and rigidity results.
result If the nextth eigenvalue is close to a lower bound, the n-soliton must be the trivial Gaussian soliton. In this short note, using Günther's volume comparison theorem and Yokota's gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton (Mn,g,f) with sectional curvature K(g)<A and Volf(M)≥v for some uniform constant A,v, there exists…
The study examines constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.
problem Characterizing constant weighted mean curvature hypersurfaces in shrinking Ricci solitons.
method Analyzing properties of hypersurfaces in specific ambient spaces (shrinking Ricci solitons).
result Conditions for a constant weighted mean curvature hypersurface to be a level set of the potential function.
We derive a local Gaussian upper bound for the f-heat kernel on complete smooth metric measure space (M,g,e−fdv) with nonnegative Bakry-Émery Ricci curvature, which generalizes the classic Li-Yau estimate. As applications, we obtain a sharp Lf1-Liouville theorem for f-subharmonic functions and an Lf1-u…
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 …
The paper estimates eigenvalues for specific differential operators on curved spaces.
problem Estimating eigenvalues for a class of elliptic differential operators on Riemannian manifolds.
method Analyzes eigenvalue estimates for a broader class of elliptic differential operators in divergence form.
result Provides eigenvalue estimates for Gaussian shrinking solitons and specific domains.
New findings on shrinking Ricci solitons with vanishing Bach-like tensors.
problem Characterizing gradient shrinking Ricci solitons with vanishing Bach-like tensors.
method Defining and analyzing Bach-like tensors, proving rigidity results, and deriving variational formulas.
result Vanishing Bach-like tensors force solitons to be either Einstein or isometric to the Gaussian soliton.
Researchers prove rigidity for log-Sobolev inequality on specific metric spaces.
problem Proving rigidity for the logarithmic Sobolev inequality on metric measure spaces.
method Using a new approach to prove the rigidity result.
result Proved that if equality holds in the log-Sobolev inequality, the space must split into a product of a manifold and the Gaussian shrinking soliton.
In this announcement, we exhibit the second variation of Perelman's λ and ν functionals for the Ricci flow, and investigate the linear stability of examples. We also define the "central density" of a shrinking Ricci soliton and compute its values for certain examples in dimension 4. Using these tools, one can somet…
In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if…
The paper classifies a type of solitons in Euclidean spaces.
problem Classifying generalized Yamabe solitons on hypersurfaces.
method Completely classified solitons arising from the position vector field.
result Classification of generalized Yamabe solitons on hypersurfaces in Euclidean spaces.
Study on shrinking solitons of generalized Ricci flow.
problem Characterizing shrinking solitons in generalized Ricci flow.
method Analyzing gradient shrinking solitons and pluriclosed solitons on compact manifolds.
result First non-trivial shrinking generalized soliton constructed.
New examples of solitons found as warped products.
problem Constructing new soliton examples.
method Warped products and explicit descriptions using elementary functions.
result Complete examples of Ricci almost solitons and Ricci-Bourguignon solitons.
Study on Yamabe solitons with applications and structure elucidation.
problem Understanding the structure of Yamabe solitons and their applications.
method Investigation of complete gradient conformal solitons under specific conditions.
result Affirmative partial answer to Yamabe soliton conjecture.
We classify Algebraic Ricci Solitons of three-dimensional Lorentzian Lie groups. All algebraic Ricci solitons that we obtain are sol-solitons. In particular, we prove that, contrary to the Riemannian case, Lorentzian Ricci solitons need not to be algebraic Ricci solitons. We classify Algebraic Ricci Solitons of three-d…
The study examines Riemann solitons and almost solitons on specific Kenmotsu manifolds.
problem Characterizing solitons on Kenmotsu manifolds.
method Analysis of Riemann solitons and gradient almost Riemann solitons on almost Kenmotsu manifolds.
result Construction of examples of Kenmotsu and (κ,μ)′-almost Kenmotsu manifolds. Study on η-Ricci-Yamabe solitons on Riemannian submersions.
problem Characterizing η-Ricci-Yamabe solitons on Riemannian submersions. method Analyzing conditions for η-Ricci-Yamabe solitons on submersions and deriving Laplacian equations. result Classification of fiber and target manifolds as η-Ricci-Yamabe solitons under various conditions. Study on geometric properties of second Ricci solitons.
problem Understanding the geometry of second Ricci solitons.
method Investigation of closed and compact second Ricci soliton manifolds, and immersed submanifolds as well as warped product manifolds.
result Investigation of geometric properties of second Ricci solitons.
Study on Ricci-like solitons and gradient solitons on specific manifolds.
problem Characterizing solitons on Sasaki-like almost contact B-metric manifolds.
method Introduced and studied Ricci-like solitons with arbitrary potential and gradient solitons. Proved properties of the Ricci tensor and soliton coefficients.
result Gradient almost Ricci-like solitons have constant soliton coefficients.
The study classifies steady Ricci solitons based on geometric conditions.
problem Characterizing steady Ricci solitons under specific geometric constraints.
method Analyzing geometric conditions and applying them to classify solitons.
result Steady Ricci solitons are classified into specific types based on given conditions.
Study on p-biharmonic maps from gradient Ricci solitons, focusing on 2D cigar soliton.
problem Understanding p-biharmonic maps on gradient Ricci solitons.
method Analyzing p-biharmonic maps from gradient Ricci solitons, specifically 2D cigar soliton.
result Obtained results on p-biharmonic maps from gradient Ricci solitons, particularly on 2D cigar soliton.
The study characterizes spacetimes with specific solitons in f(R)-gravity.
problem Characterizing spacetimes with specific solitons in f(R)-gravity. method Analyzing η-Ricci solitons, gradient η-Ricci solitons, gradient Einstein Solitons, and gradient m-quasi Einstein solitons in perfect fluid spacetimes obeying f(R)-gravity. result Established conditions for the behavior of η-Ricci solitons and derived significant theorems about dark matter. We prove some results for the solitons of the Ricci-Bourguignon flow, generalizing corresponding results for Ricci solitons. Taking motivation from Ricci almost solitons, we then introduce the notion of Ricci-Bourguignon almost solitons and prove some results about them which generalize previous results for Ricci alm…
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…
Extends soliton theory to non-compact cases.
problem Generalized solitons in non-compact settings.
method Place conditions on vector field and curvature, use tensor properties.
result Non-compact q-solitons are stationary and q-flat. Paper shows constant σk-curvature for quasi k-Yamabe solitons.
problem Understanding constant curvature in quasi k-Yamabe solitons.
method Analyzes conditions for solitons to be gradient and constant curvature.
result Compact quasi k-Yamabe solitons have constant σk-curvature.
Proves uniqueness of solutions for a nonlocal Liouville equation with finite Q-curvature.
problem Proving uniqueness of solutions for a specific nonlocal Liouville equation.
method Connection to Calogero--Moser derivative NLS and ground state solitons.
result Uniqueness of solutions in the Gaussian case and general positive, symmetric-decreasing K. Researchers study solitons on homogeneous manifolds, proving properties of specific types of solitons.
problem Examining solitons on homogeneous manifolds to understand their properties and constraints.
method Analyzing ambient obstruction flow and specific solitons in homogeneous spaces, proving properties and constructing examples.
result Proved that any compact ambient obstruction soliton with constant scalar curvature is trivial, and characterized specific types of solitons in 4-dimensional homogeneous spaces.
The paper characterizes Ricci solitons on the Poincaré upper half plane.
problem Characterizing Ricci solitons on the Poincaré upper half plane.
method Classifying and generalizing Ricci solitons and soliton equations in the half plane of Poincaré.
result Obtained some properties of solitons about their geodesic flows.
Characterizes ∗-k-Ricci-Yamabe solitons on Kenmotsu manifolds.
problem Understanding ∗-k-Ricci-Yamabe solitons on Kenmotsu manifolds. method Analyzes the geometry of ∗-k-Ricci-Yamabe solitons and gradient solitons on Kenmotsu manifolds. result Characterizes the nature of ∗-k-Ricci-Yamabe solitons and gradient solitons. The paper explores almost Ricci solitons on Finsler spaces, proving conditions for their existence.
problem Characterizing almost Ricci solitons on Finsler measure spaces.
method Introducing and investigating gradient almost Ricci solitons, proving conditions for existence.
result Conditions for the existence of gradient almost Ricci solitons on Finsler measure spaces.