The paper studies how certain solitons on Fano manifolds extend to nearby deformations.
problem Conditions for weighted solitons to extend to nearby deformations of Fano manifolds.
method Analyzes the Kuranishi family of Fano manifolds and uses equivariant automorphism groups.
result All members of the Kuranishi family of a Fano manifold with a weighted soliton have weighted solitons if and only if the dimensions of their T-equivariant automorphism groups are equal to that of the original manifold.
This paper connects soliton-type metrics with weighted CSCK metrics on Fano manifolds.
problem Existence and properties of weighted constant scalar curvature Kähler metrics.
method Introducing a weight function g(v,w) and proving equivalence between (v,w)-CSCK metrics and g(v,w)-solitons. result Existence of (v,w)-CSCK metrics in the first Chern class is equivalent to existence of g(v,w)-solitons. 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.
Survey on Kähler-Einstein and weighted solitons on Fano manifolds.
problem Existence of coupled Kähler-Einstein metrics and weighted solitons on Fano manifolds.
method Generalization of algebraic conditions for K-polystability.
result Existence of coupled Kähler-Einstein metrics and weighted solitons is equivalent to algebraic conditions.
The paper proves inequalities for hypersurfaces in weighted manifolds.
problem Willmore-type inequalities for closed hypersurfaces in weighted manifolds.
method Analyzes weighted manifolds with nonnegative Bakry-Émery Ricci curvature, proving sharp inequalities and characterizing equality cases.
result Derives sharp Willmore-type and Willmore-like inequalities in steady and shrinking gradient Ricci solitons.
Study shows instability of Kähler Ricci solitons and stability of orbifold singularities.
problem Linear stability and instability of Kähler Ricci solitons.
method Extending the approach of \cite{chi04} and \cite{hm11}, via recent work \cite{cm21} on gradient shrinking Ricci solitons.
result Linear instability of the BCCD shrinking soliton and stability of orbifold singularities of Kähler solitons.
Study on Einstein solitons with bounds and asymptotic behavior.
problem Understanding the properties of Einstein solitons.
method Computed lower bounds for scalar curvature, established asymptotic behavior, proved finiteness of fundamental group and weighted volume.
result Established finiteness of fundamental group and weighted volume for gradient shrinking Einstein solitons.
The article characterizes gradient ρ-Einstein solitons under specific conditions.
problem Characterizing gradient ρ-Einstein solitons with certain properties.
method Analyzing solitons with vector fields of bounded norm, finite weighted Dirichlet integral, and specific Ricci curvature restrictions.
result Non-trivial complete gradient ρ-Einstein solitons with finite weighted Dirichlet integral and certain Ricci curvature restrictions are of constant scalar curvature and steady.
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.
New examples of Kähler-Ricci solitons on Fano threefolds with non-trivial moduli found.
problem Finding Fano threefolds with Kähler-Ricci solitons and non-trivial moduli.
method Established weighted K-stability and GIT-stability criteria, generalized Koiso's theorem, and developed the weighted Abban-Zhuang estimate.
result First examples of strictly weighted K-semistable Fano varieties and new examples of KRS Fano varieties with non-trivial moduli and small automorphism groups.
Gradient Ricci solitons can be extended to non-gradient Ricci solitons using energy function.
problem Extending the geometry of gradient Ricci solitons to non-gradient Ricci solitons.
method Using energy function E to study the geometry. result A non-steady Ricci soliton with symmetric covariant derivative is gradient.
We introduce a natural extension of the concept of gradient Ricci soliton: the Ricci almost soliton. We provide existence and rigidity results, we deduce a-priori curvature estimates and isolation phenomena, and we investigate some topological properties. A number of differential identities involving the relevant geome…
New insights into Kähler Ricci solitons and Calabi-Yau cones.
problem Understanding Kähler Ricci solitons and their relationship to Calabi-Yau cones.
method Analyzing the canonical cone of Fano manifolds and using openness of weight functions.
result The canonical cone of a product of a smooth Fano manifold and a complex projective space is a Calabi-Yau cone under certain conditions.
Exposes new connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.
problem Exploring connections between weighted Kähler-Ricci solitons and Ricci-flat Kähler cone metrics.
method Surveying and reviewing recent works, transforming complex Monge-Ampère equations, and proving the YTD conjecture.
result Established a transformation from irregular Sasaki-Einstein metrics to g-solitons on quasi-regular quotients. The study classifies translating and self-expanding solitons in 3D space.
problem Characterizing the topology and index of solitons in mean curvature flow.
method Analyzing the spectrum and index of expanding and translating solitons in R3. result Translating and self-expanding solitons have finite topology under certain conditions.
In this paper, we prove that a noncompact complete hypersurface with finite weighted volume, weighted mean curvature vector bounded in norm, and isometrically immersed in a complete weighted manifold is proper. In addition, we obtain an estimate for f-stability index of a constant weighted mean curvature hypersurface…
The paper proves instability of translating λ-solitons and provides bounds on their length.
problem Stability of translating λ-solitons in cylindrical geometry.
method Analytical proof of instability and explicit length bounds.
result Explicit bounds on the length of unstable translating λ-solitons.
The study shows ends of shrinking gradient ρ-Einstein solitons are non-parabolic.
problem Characterizing the ends of shrinking gradient ρ-Einstein solitons. method Proving non-parabolicity of ends and connectivity at infinity for specific conditions.
result Gradient shrinking ρ-Einstein solitons have non-parabolic ends under certain conditions. We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted $\barpartial$-Laplacian on compact Kähler manifolds. This situation typically occurs for a sequence of Fano manifolds with anticanonical Kähler class. We apply it to show that, if an almost smooth Fano-Ricci limit space admits a Kähler…
The paper characterizes solitons and estimates scalar curvature.
problem Characterizing and estimating scalar curvature of generalized Ricci-Yamabe solitons.
method Characterization and estimation of scalar curvature through soliton properties.
result Conditions for scalar curvature to be constant and estimation of Ricci curvature.
Investigates noncompact warped product Ricci solitons, proving nonexistence results.
problem Proving nonexistence results for noncompact warped product Ricci solitons.
method Proves nonexistence results using pde estimates and properties of the first eigenvalue of a weighted Laplacian.
result Nonexistence theorem for shrinking Ricci solitons.
The paper proves conditions for Einstein solitons to split into line and manifold.
problem Conditions for Einstein solitons to split into line and manifold.
method Weighted Laplacian comparison of distance function and bounded integral condition on Ricci curvature.
result Gradient ρ-Einstein solitons split off a line isometrically under certain conditions.
Proves conditions for weighted Hermite-Einstein metrics on vector bundles.
problem Conditions for existence of weighted Hermite-Einstein metrics.
method Introduces weighted Hermite-Einstein equation, stability notions, and proves existence.
result Existence of weighted Hermite-Einstein metrics if and only if slope polystable.
Paper proves stability and Dirichlet problem for translating hypersurfaces.
problem Stability and Dirichlet problem for translating hypersurfaces.
method Analyzes translating solitons in en+k, proves stability conditions, and studies Dirichlet problem. result Proves the infimum of mean curvature is zero for translating solitons and conditions for stability.
Study solves Yamabe problems on metric measure spaces with or without boundary.
problem Yamabe-type problems on compact metric measure spaces with or without boundary.
method Analyzes uniqueness, characterization, and existence of minimizers.
result Characterizes weighted Yamabe solitons and existence of positive minimizers.
New stable shrinking Ricci soliton found in 4D.
problem Stability of 4D Ricci shrinkers.
method Explicit metric description, tensor harmonic analysis, Kähler structure, selfduality insights.
result First non-cylindrical linearly stable shrinking Ricci soliton in 4D.
Study on 4D solitons with curvature constraints.
problem Characterizing gradient shrinking Ricci solitons with positive modified sectional curvature.
method Sharp pinching conditions, weighted integral gap results, Hitchin-Thorpe inequality.
result Locally Kähler property under specific curvature conditions.
We show that the generalized Kähler-Ricci soliton equation on 4-dimensional toric Kähler orbifolds reduces to ODEs assuming there is a Hamiltonian 2-form. This leads to an explicit resolution of this equation on labeled triangles and convex labeled quadrilaterals. In particular, we give the explicit expression of the K…
Study geometric and analytical properties of ρ-Einstein solitons.
problem Characterize geometric and analytical features of ρ-Einstein solitons. method Analyze the spectrum of the drifted Laplacian operator and prove volume growth estimates.
result Establish new volume growth estimates for geodesic balls of complete noncompact ρ-Einstein solitons. In this paper we consider the Ricci curvature of a Ricci soliton. In particular, we have showed that a complete gradient Ricci soliton with non-negative Ricci curvature possessing a non-constant convex potential function having finite weighted Dirichlet integral satisfying an integral condition is Ricci flat and also i…
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.
New invariants help solve existence of weighted cscK metrics.
problem Existence of weighted cscK metrics in K-stability.
method Introduced weighted analytic delta invariant and beta invariant.
result Sufficient condition for existence of weighted cscK metrics.
Study shows only grim reaper cylinder for certain self-translating surfaces.
problem Characterizing self-translating surfaces in 3D space.
method Used parabolicity in a weighted setting and universally L-superharmonic functions.
result Characterized the grim reaper cylinder as the only finite entropy self-translating 2-surface in R^3 of width π and bounded from below.
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…
We study both function theoretic and spectral properties of the weighted Laplacian Δf on complete smooth metric measure space (M,g,e−fdv) with its Bakry-Émery curvature Ricf bounded from below by a constant. In particular, we establish a gradient estimate for positive f−harmonic functions and a sharp upper…
Proves existence of weighted-cscK metrics on Kähler manifolds.
problem Existence of weighted-cscK metrics on Kähler manifolds.
method Proves existence via G-coercivity of weighted Mabuchi functional. result Existence of (v, w)-weighted-cscK metrics with v log-concave.
In this paper we study solitons invariant with respect to the flow generated by a complete Killing vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product (R×P,dt2+g0) and the Killing field is X=∂t. Similarly to what h…
The paper proves rigidity and vanishing theorems for translating solitons.
problem Understanding the properties of translating solitons in geometry.
method Using Sobolev inequalities and Lq-norms, the paper proves rigidity and vanishing theorems. result Translating solitons are shown to be hypersurfaces under certain conditions.
We consider dynamical stability for a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics. Our focus is on homogeneous metrics on non-compact manifolds. Following the program of Guenther, Isenberg, and Knopf, we define a class of weighted little Hölder spaces with certain …
In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and Lp-Liouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scal…
Paper explores new Kähler metrics from old, aiming to solve YTD conjecture.
problem Extending classical extremal Kähler metrics to include new objects.
method Surveying recent works on weighted extremal Kähler metrics and the YTD conjecture.
result Survey of recent research on weighted extremal Kähler metrics.
In this paper we give some results on the topology of manifolds with ∞-Bakry-Émery Ricci tensor bounded below, and in particular of steady and expanding gradient Ricci solitons. To this aim we clarify and further develop the theory of f-harmonic maps from non-compact manifolds into non-positively curved manifold…
As a generalization of Kahler-Einstein metrics for Fano manifolds with nonvanishing Futaki invariant, Mabuchi solitons are critical points of a Calabi-type energy functional. We study their existence on toric Fano varieties and the underlying algebraic stability notion: relative Ding stability. As a toy model for a YTD…
In this paper we show that an immersed nontrivial translating soliton for mean curvature flow in Rn+1(n=2,3) is a grim hyperplane if and only if it is mean convex and has weighted total extrinsic curvature of at most quadratic growth. For an embedded translating soliton Σ with nonnegative scalar curva…
We generalize a method by L. Ambrozio, A. Carlotto, and B. Sharp to study the Morse index of closed f-minimal hypersurfaces isometrically immersed in a general weighted manifold. The technique permits, in particular, to obtain a linear lower bound on the Morse index via the first Betti number for closed f-minimal hyper…
We use the weighted Hsiung-Minkowski integral formulas and Brendle's inequality to show new rigidity results. First, we prove Alexandrov type results for closed embedded hypersurfaces with radially symmetric higher order mean curvature in a large class of Riemannian warped product manifolds, including the Schwarzschild…
Blowups of Kähler manifolds can inherit extremal metrics.
problem Extending extremal metrics to blowups of Kähler manifolds.
method Analyzing the action of a torus on blowups and weighted extremal metrics.
result Blowups of Kähler manifolds can inherit weighted extremal metrics.
In this paper, we study Lichnerowicz type estimate for eigenvalues of drifting Laplacian operator and L1 and L2 energy for drifting heat equation on closed manifolds with weighted measure. In some sense, this study is about the eigenvalue estimate on Ricci solitons.