arXiv research

Researchers create a family of solitons connecting a cigar to a sphere.

The only known example of collapsed three-dimensional complete gradient steady Ricci solitons so far is the 3D cigar soliton $N^2\times \mathbb{R}$, the product of Hamilton's cigar soliton $N^2$ and the real line $\mathbb{R}$ with the product metric. R. Hamilton has conjectured that there should exist a family of colla…

Compact Ricci solitons on surfaces have at most two cone points, and are known as Hamilton's footballs. In this note we completely describe the degenerations of these footballs as one or both of the cone angles approaches zero. In particular, we show that Hamilton's famous non-compact cigar soliton is the Gromov--Hausd…

Study on mixed Killing vector fields on Cigar Ricci-Bourguignon solitons.

We produce new non-Kähler complete steady gradient Ricci solitons whose asymptotics combine those of the Bryant solitons and the Hamilton cigar. We also obtain a family of complete Ricci-flat metrics with asymptotically locally conical asymptotics. Finally, we obtain numerical evidence for complete steady soliton struc…

Let (M,g) be a Riemannian manifold with an isometric action of the Lie group G. Let g_G be a left invariant metric on G. Consider the diagonal G action on the product $M \times G$ with the metric g+g_G. In this paper we calculate the formula for the metric h on the quotient space $(M \times G) / G$; the map from g to h…

Study on p-biharmonic maps from gradient Ricci solitons, focusing on 2D cigar 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 study sequences of 3-dimensional solutions to the Ricci flow with almost nonnegative sectional curvatures and diameters tending to infinity. Such sequences may arise from the limits of dilations about singularities of Type IIb. In particular, we study the case when the sequence collapses, which may occur when dilati…

In this paper we consider a perturbation of the Ricci solitons equation proposed by J. P. Bourguignon in \cite{jpb1}. We show that these structures are more rigid then standard Ricci solitons. In particular, we prove that there is only one complete three--dimensional, positively curved, Riemannian manifold satisfying $…

We study the non Ricci flat gradient steady Kähler Ricci soliton with non-negative Ricci curvature and weak integrability condition of the scalar curvature $S$, namely $\underline{\lim}_{r\to \infty} r^{-1}\int_{B_r} S=0$, and show that it is a quotient of $Σ\times \mathbb{C}^{n-1-k}\times N^k$, where $Σ$ and $N$ denot…

We show that the Cigar metric on $\mathbb{C}$ is an example of real analytic Kähler manifold with globally defined and positive Calabi's diastasis function which cannot be Kähler immersed into any (finite or infinite dimensional) complex space form.

In this paper, we study the question if there is an isometric immersion of the cigar soliton into $R^3$. We show that the answer is negative. Similar result in higher dimensions is also true for steady Bryant solitons.

Analyzes harmonic and biharmonic maps from gradient Ricci solitons.

In this paper, we shall use the Kähler geometry formulation to study the global behavior of the Ricci flow on $R^2$. The geometric feature of our Ricci flow is that it has finite width. Our aim is to determine the limiting metric (which corresponds an eternal Ricci flow) obtained by L.F.Wu. We can use the classificatio…

Study on stellar models' topology and mass using minimal surfaces.

This is an exposition of aspects of the result of Daskalopoulos and Sesum that any 2-dimensional complete noncompact ancient solution to Ricci flow with bounded positive scalar curvature and finite width must be the cigar soliton.

The study classifies steady Ricci solitons based on geometric conditions.

We characterize complete nonnegatively curved steady gradient soliton with curvature in L^1. We show that there are isometric to a product (R^2,g_{cigar}) times(R^{n-2}, eucl))/Gamma where Gamma is a Bieberbach group of rank n-2. We prove also a similar local splitting result under weaker curvature assumptions.

Type II (ancient) solutions to the Ricci flow on surfaces are not yet classified. It is conjectured that the Rosenau solution and the cigar are the only solutions, modulo scaling. In this paper, we mainly study the backward limit and the circumference at spatial infinity of Type II ancient solutions on noncompact surfa…

We prove that the isoperimetric inequality is satisfied in the cigar steady soliton and in the Bryant steady soliton. Since both of them are Riemannian manifolds with warped product metric, we utilize the result of Guan-Li-Wang to get our conclusion. For the sake of the soliton structure, we believe that the geometric …

Paper proves Hamilton's pinching theorem using mean curvature flow.

Proves Hamilton's theorem using mean curvature flow.

Reduction theory has played a major role in the study of Hamiltonian systems. On the other hand, the Hamilton-Jacobi theory is one of the main tools to integrate the dynamics of certain Hamiltonian problems and a topic of research on its own. Moreover, the construction of several symplectic integrators rely on approxim…

Diffieties formalize geometrically the concept of differential equations. We introduce and study Hamilton-Jacobi diffieties. They are finite dimensional subdiffieties of a given diffiety and appear to play a special role in the field theoretic version of the geometric Hamilton-Jacobi theory.

Survey reviews Hamilton-Jacobi theory in various geometric settings, focusing on Jacobi and Leibniz identities.

In this paper we develop a Hamilton-Jacobi theory in the setting of almost Poisson manifolds. The theory extends the classical Hamilton-Jacobi theory and can be also applied to very general situations including nonholonomic mechanical systems and time dependent systems with external forces.

Develops Lagrange-Hamilton geometry for COVID-19 disease dynamics.

This paper provides a geometric description for Lie--Hamilton systems on $\mathbb{R}^2$ with locally transitive Vessiot--Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie--Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves re…

New graph Hamiltonicity via cohomology of Artin groups.

Paper proves Hamilton-Tian conjecture using partial C0-estimate.

Paper confirms Hamilton-Tian conjecture for specific Sasakian manifolds.

The first widely used financial model is linked to dynamical Hamilton jacobi model

Proves estimates for Kähler-Ricci flow solutions.

This paper studies Hamilton-Jacobi equations for magnetic systems with constraints.

The Hamilton-Jacobi equation for a Hamiltonian section on a Lie affgebroid is introduced and some examples are discussed.

I briefly review my proposal about how to extend the geometric Hamilton-Jacobi theory to higher derivative field theories on fiber bundles.

In this paper, we provide an essentially self-contained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of three-manifolds. In particular, we give a detailed exposition of a complete proof of the Poincaré con…

The article proves a new entropy formula for surfaces with boundaries.

This paper has been withdrawn by the author due to the version of [A complete proof of Hamilton's conjecture] at arXiv:1008.1576

The interplay between the Hamilton-Jacobi theory of orthogonal separation of variables and the theory of group actions is investigated based on concrete examples.

We derive an interpolation version of constrained matrix Li-Yau-Hamilton estimate on Kähler manifolds. As a result, we first get a constrained matrix Li-Yau-Hamilton estimate for heat equation on a Kähler manifold with fixed Kähler metric. Secondly, we get a corresponding estimate for forward conjugate heat equation on…

Kuranishi's proof of complex deformation theory revisited

In this paper, we will recover Hamilton's Harnack inequality for the Ricci flow from the view point of Hyperbolic thermostat.

We give a brief survey of Hamilton's program for 3-manifolds as an approach toward Thurston's Geometrization Conjectre.

The paper proves estimates for a specific flow on compact manifolds.

There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the Abelian vortex equations, and it is shown that the property that a metric solve t…

In this paper, we give precisely the geometric constraint conditions of canonical symplectic form and regular reduced symplectic forms for the dynamical vector fields of a regular controlled Hamiltonian (RCH) system and its regular reduced systems, which are called the Type I and Type II of Hamilton-Jacobi equations. A…