arXiv research

Study on geometric flows and rigidity of solitons.

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

We use the exterior product of double forms to reformulate celebrated classical results of linear algebra about matrices and bilinear forms namely the Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton identities and Jacobi's formula for the determinant. This new formalism is then used to naturally g…

Two new classes of metrizable vector bundles have been presented in the papers [1] and [4]. The Lie algebroid generalized tangent bundle of a dual vector bundle is presented. This Lie algebroid is a new example of metrizable vector bundle. A new class of Hamilton spaces, called by use, generalized Hamilton (ρ,η)-space,…

We introduce the new notion of Bianchi-convex sets, a generalization of convex sets of algebraic curvature tensors inspired by the second Bianchi identity. It turns out that Hamilton's maximum principle for the Ricci flow can be generalized for Bianchi-convex sets.

In this paper, we study the following conjecture of Hamilton: Any compact gradient shrinking Ricci soliton with positive curvature operator must be Einstein. We first derive several identities. Then we show that the conjecture is true under an additional condition. Furthermore, such a soliton must be of constant curvat…

We show that classical thermodynamics has a formulation in terms of Hamilton-Jacobi theory, analogous to mechanics. Even though the thermodynamic variables come in conjugate pairs such as pressure/volume or temperature/entropy, the phase space is odd-dimensional. For a system with n thermodynamic degrees of freedom it …

The closed string model in the background gravity field is considered as a bi-Hamiltonian system in assumption that string model is the integrable model for particular kind of the background fields. The dual nonlocal Poisson brackets(PB), depending of the background fields and of their derivatives, are obtained. The in…

Study of gyroscopic Chaplygin systems and magnetic flows on spheres.

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

This paper optimizes reinsurance contracts with belief differences between insurer and reinsurer.

We study the structure of a simple dynamic optimization problem consisting of one state and one control variable, from a physicist's point of view. By using an analogy to a physical model, we study this system in the classical and quantum frameworks. Classically, the dynamic optimization problem is equivalent to a clas…

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…

We study power expansions of the characteristic function of a linear operator $A$ in a $p|q$-dimensional superspace $V$. We show that traces of exterior powers of $A$ satisfy universal recurrence relations of period $q$. `Underlying' recurrence relations hold in the Grothendieck ring of representations of $\GL(V)$. The…

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…

Study optimal portfolio strategy with sporadic bankruptcy for isoelastic utility.

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.

A few generalizations of a Poisson algebra to field theory canonically formulated in terms of the polymomentum variables are discussed. A graded Poisson bracket on differential forms and an $(n+1)$-ary bracket on functions are considered. The Poisson bracket on differential forms gives rise to various generalizations o…

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 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.

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.

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

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.

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

In this paper, by employ the cutoff function and the maximum principle, some Hamilton-Souplet-Zhang type gradient estimates for porous medium type equation are deduced. As a special case, an Hamilton-Souplet-Zhang type gradient estimates of the heat equation is derived which is different from the result of Souplet-Zhan…

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…

Alternative proof of flatness for Ricci-pinched 3-manifolds.