arXiv research

New graph Hamiltonicity via cohomology of Artin groups.

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…

Proves estimates for Kähler-Ricci flow solutions.

Paper proves inequality for Green function on Kähler manifolds.

Improved heat equation estimates without gradient curvature assumption.

We proved a matrix Li-Yau-Hamilton type gradient estimates for the positive solutin of the heat equation on complete Kaehler manifolds with nonnegative bisectional curvature. As a consequence we obtain a comparison theorem for the distance function under this curvature assumption.

We give a geometric interpretation of Hamilton's matrix Harnack inequality for the Ricci flow as the curvature of a connection on space-time.

Paper proves estimates for heat and conjugate heat equations under Ricci flow, leading to monotonicity of parabolic frequencies.

We generalize Hamilton's matrix Li-Yau-type Harnack estimate for the Ricci flow by considering the space of all LYH (Li-Yau-Hamilton) quadratics that arise as curvature tensors of space-time connections satisfying the Ricci flow with respect to the natural space-time degenerate metric. As a special case, we employ scal…

In this paper we are concerned with the matrix Li-Yau-Hamilton estimates for nonlinear heat equations. Firstly, we derive such estimate on a Kähler manifold with a fixed Kähler metric. Then we consider the estimate on Kähler manifolds with Kähler metrics evolving under the rescaled Kähler-Ricci flow. Both of the estima…

Based on a suggestion of Richard Hamilton, we give an alternate proof of his matrix Harnack inequality for solutions of the Ricci flow with positive curvature operator. This Harnack inequality says that a certain endomorphism, consisting of an expression in the curvature and its first two covariant derivatives, of the …

Derives matrix Harnack inequalities for semilinear heat equations on manifolds.

In this paper we study the heat equation (of Hodge-Laplacian) deformation of $(p, p)$-forms on a Kähler manifold. After identifying the condition and establishing that the positivity of a $(p, p)$-form solution is preserved under such an invariant condition we prove the sharp differential Harnack (in the sense of Li-Ya…

In this paper we prove a new matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow. The form of this new Li-Yau-Hamilton estimate is obtained by the interpolation consideration originated in \cite{Ch1}. This new inequality is shown to be connected with Perelman's entropy formula through a family of differential equalit…

Study of Hamilton-Jacobi Theory with symmetries and integrability by quadratures.

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

Proves Hamilton's theorem using mean curvature flow.

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…

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…

This work is devoted to the study of parabolic frequency for solutions of the heat equation on Riemannian manifolds. We show that the parabolic frequency functional is almost increasing on compact manifolds with nonnegative sectional curvature, which generalizes a monotonicity result proved by C. Poon and by L. Ni. The…

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

Paper confirms Hamilton-Tian conjecture for specific Sasakian manifolds.

The paper develops RL methods for optimal switching between multiple states.

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

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.

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.

Study on singularities of solutions to Hamilton-Jacobi equations on manifolds.

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…

Paper introduces stochastic HJB on Jacobi structures.

In this paper, we study the gradient estimates of Li-Yau-Hamilton type for positive solutions to both drifting heat equation and the simple nonlinear heat equation problem $u_t-Δu=au\log u, \ \ u>0$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative (Bakry-Emery)-Ricci curvature. Here…

Study magnetic Hamiltonian systems with constraints, deriving Hamilton-Jacobi equations.

Develops Hamilton-Jacobi theory for non-conservative field theories in k-contact geometry.

Study on geometric flows and rigidity of solitons.

Proves algebraic version of Hamilton-Tian conjecture for log Fano pairs.