arXiv research

New gravitational solitons and infinite topological manifolds found.

Undergraduate thesis explores topological barriers to compact Ricci solitons in 4D.

The paper examines the geometry and topology of Sasaki-Ricci solitons, proving they are either connected at infinity or compact.

In the present article we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow.

The study classifies translating and self-expanding solitons in 3D space.

In this paper, we prove that a sequence of weak almost Kähler-Ricci solitons under further suitable conditions converge to a Kähler-Ricci soliton with complex codimension of singularities at least 2 in the Gromov-Hausdorff topology. As a corollary, we show that on a Fano manifold with the modified K-energy bounded belo…

Study cohomogeneity one expanding Ricci solitons on specific topologies.

We prove that any shrinking Kahler Ricci soliton has only one end, and that any expanding Kahler Ricci soliton with proper potential has only one end.

We construct an expanding gradient Ricci soliton in dimension three over the topological manifold R x T^2 (the product of a line and a torus) that aproaches asymptotically a constant curvature cusp at one end, and a flat manifold on the other end. We prove that this is the only gradient soliton with this topology, prov…

New steady gradient Ricci solitons found on specific four-manifolds.

Non-trivial obstructions found for topological solitons in Yang-Mills-Chern-Simons theories.

Study of curvature flow on complex Lie groups, leading to soliton convergence.

In this paper we obtain rigidity results and obstructions on the topology at infinity of translating solitons of the mean curvature flow in the Euclidean space. Our approach relies on the theory of f-minimal hypersurfaces.

We show that any complete $f$-stable translating soliton $M$ admits no codimension one cycle which does not disconnect $M$. As a corollary, it follows that any two dimensional complete $f$-stable translating soliton has genus zero.

In this article we prove the existence of Kahler-Ricci solitons on smoothable, K-stable Q-Fano varieties. We also investigate the behavior of twisted Kahler-Ricci solitons in the Gromov-Hausdorff topology under this smoothing family.

In this note, we complete the classification of the geometry of non-compact two-dimensional gradient Ricci solitons. As a consequence, we obtain two corollaries: First, a complete two-dimensional gradient Ricci soliton has bounded curvature. Second, we give examples of complete two-dimensional expanding Ricci solitons …

We show that the topological charge of the n-soliton solution of the sine-Gordon equation n is related to the genus g > 1 of a constant negative curvature compact surface described by this configuration. The relation is n=2(g-1), where n is even. The moduli space of complex dimension B(g)=3(g-1) corresponds precisely t…

In this paper, we use less topological restrictions and more geometric and analytic conditions to obtain some sufficient conditions on Yamabe solitons such that their metrics are Yamabe metrics, that is, metrics of constant scalar curvature. More precisely, we use properties of conformal vector fields to find several s…

We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its Bakry-Émery Ricci tensor has a positive lower bound, and either of the following conditions: (i) the Ricci curvature is bounded from above; (ii) the Ricci curvat…

We derive lower bounds on the scalar curvature of complete non-compact gradient Yamabe solitons under some integral curvature conditions. Based on this, we prove that the corresponding potential functions have at most quadratic growth in distance. We also obtain a finite topological type property on complete shrinking …

In this note we discuss estimates for the curvature of 4-dimensional gradient Ricci soliton singularity models by applying Perelman's point selection, a fundamental result of Cheeger and Naber, and topological lemmas.

Study on 4D solitons with curvature constraints.

In this paper we consider connections between Ricci solitons and Einstein metrics on homogeneous spaces. We show that a semi-algebraic Ricci soliton admits an Einstein one-dimensional extension if the soliton derivation can be chosen to be normal. Using our previous work on warped product Einstein metrics, we show that…

In this short paper, we improve the result of Phong-Song-Sturm on degeneration of Fano Kähler-Ricci solitons by removing the assumption on the uniform bound of the Futaki invariant. Let $\mathcal{KR}(n)$ be the space of Kähler-Ricci solitons on $n$-dimensional Fano manifolds. We show that after passing to a subsequence…

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…

We consider a class of time dependent finite energy multi-soliton solutions of the U(N) integrable chiral model in $(2+1)$ dimensions. The corresponding extended solutions of the associated linear problem have a pole with arbitrary multiplicity in the complex plane of the spectral parameter. Restrictions of these exten…

Study of ends of complete gradient Schouten solitons, showing finitely many ends for shrinking and connected infinity for expanding ones.

In this paper we give some results on the topology of manifolds with $\infty$-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…

We construct a canonical Hausdorff complex analytic moduli space of Fano manifolds with Kähler-Ricci solitons. This naturally enlarges the moduli space of Fano manifolds with Kähler-Einstein metrics, which was constructed by Odaka and Li-Wang-Xu. We discover a moment map picture for Kähler-Ricci solitons, and give comp…

This paper derives new identities for the Weyl tensor on a gradient Ricci soliton, particularly in dimension four. First, we prove a Bochner-Weitzenböck type formula for the norm of the self-dual Weyl tensor and discuss its applications, including connections between geometry and topology. In the second part, we are co…

We consider the space KR(n,F) of Kahler-Ricci solitons on n-dimensional Fano manifolds with Futaki invariant bounded by F. We prove a partial C^0 estimate for KR(n,F) as a generalization of the recent work of Donaldson-Sun for Fano Kahler-Einstein manifolds. In particular, any sequence in KR(n,F) has a convergent sub…

The paper explores non-minimal solitons in the sphere with unique properties.

We investigate self-similar solutions to the inverse mean curvature flow in Euclidean space. In the case of one dimensional planar solitons, we explicitly classify all homothetic solitons and translators. Generalizing Andrews' theorem that circles are the only compact homothetic planar solitons, we apply the Hsiung-Min…

We construct many self-similar and translating solitons for Lagrangian mean curvature flow, including self-expanders and translating solitons with arbitrarily small oscillation on the Lagrangian angle. Our translating solitons play the same role as cigar solitons in Ricci flow, and are important in studying the regular…

Haslhofer and Müller proved a compactness Theorem for four-dimensional shrinking gradient Ricci solitons, with the only assumption being that the entropy is uniformly bounded from below. However, the limit in their result could possibly be an orbifold Ricci shrinker. In this paper we prove a compactness theorem for non…

We characterize $η$-Ricci solitons $(g,ξ,λ,μ)$ in some special cases when the $1$-form $η$, which is the $g$-dual of $ξ$, is a harmonic or a Schrödinger-Ricci harmonic form. We also provide necessary and sufficient conditions for $η$ to be a solution of the Schrödinger-Ricci equation and point out the relation between …

New degree theory proves existence of solitons on 4D manifolds.

New stable shrinking Ricci soliton found in 4D.

In dimension $n=3$, there is a complete theory of weak solutions of Ricci flow - the singular Ricci flows introduced by Kleiner and Lott - which are unique across singularities, as was proved by Bamler and Kleiner. We show that uniqueness should not be expected to hold for Ricci flow weak solutions in dimensions $n\geq…

New framework models high-Hopf-index hopfions using generalized fold maps.

The stability of physical systems depends on the existence of a state of least energy. In gravity, this is guaranteed by the positive energy theorem. For topological reasons this fails for nonsupersymmetric Kaluza-Klein compactifications, which can decay to arbitrarily negative energy. For related reasons, this also fa…

Study $\ast$-$η$-Ricci solitons on weak Kenmotsu $f$-manifolds.

Study Ricci flow on spaces with conical singularities, proving existence and curvature estimates.

Study of Heterotic solitons on 4-manifolds, focusing on their properties and deformations.

We study both function theoretic and spectral properties of the weighted Laplacian $Δ_f$ on complete smooth metric measure space $(M,g,e^{-f}dv)$ with its Bakry-Émery curvature $Ric_f$ bounded from below by a constant. In particular, we establish a gradient estimate for positive $f-$harmonic functions and a sharp upper…

In 1967, Japanese physicist Morikazu Toda published the seminal papers exhibiting soliton solutions to a chain of particles with nonlinear interactions between nearest neighbors. In the decades that followed, Toda's system of particles has been generalized in different directions, each with its own analytic, geometric,…

Study of curvature flow on specific Lie groups, leading to soliton solutions.

Study of $η$-Ricci solitons and $η$-Einstein metrics on weak $β$-Kenmotsu $f$-manifolds.