arXiv research

We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log te…

We establish a stability result for elliptic and parabolic complex Monge-Amp{è}re equations on compact K{ä}hler manifolds, which applies in particular to the K{ä}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.

The paper constructs invariant Calabi-Yau structures on complexified symmetric spaces.

We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural ass…

The paper studies mean curvature flow of Lagrangian graphs in pseudo-Euclidean space.

Continuity of complex Monge-Ampère potentials on Kähler manifolds.

We compare various notions of weak subsolutions to degenerate complex Monge-Amp{è}re flows, showing that they all coincide. This allows us to show that the viscosity solution coincides with the envelope of pluripotential subsolutions. Dedicated to Duong Hong Phong on the occasion of his 65th birthday.

Study solves complex equation on specific types of manifolds.

We obtain a necessary and sufficient condition of existence of a K{ä}hler-Einstein metric on a $G\times G$-equivariant Fano compactification of a complex connected reductive group $G$ in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the …

We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations - the generalised Monge-Amp`ere equation, the almost Hitchin system, and the Calabi-Yang-Mills equations. These are all perturbations of already existing equations. Our construction for the g…

We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, while the Hessian is approximated by automatic differentiation of the gradient…

N. V. Efimov \cite{Ef1} proved that there is no complete, smooth surface in $\R^3$ with uniformly negative curvature. We extend this to isometric immersions in a 3-manifold with pinched curvature: if $M^3$ has sectional curvature between two constants $K_2$ and $K_3$, then there exists $K_1 < \min(K_2, 0)$ such that $M…

Solves complex Monge-Ampère equations on Kähler manifolds.

In this paper, we investigate two hyperbolic flows obtained by adding forcing terms in direction of the position vector to the hyperbolic mean curvature flows in \cite{klw,hdl}. For the first hyperbolic flow, as in \cite{klw}, by using support function, we reduce it to a hyperbolic Monge-Amp$\grave{\rm{e}}$re equation …

Study complex Monge-Ampère flows on Kähler manifolds using Perron method.

We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop K{ä}hler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin-Abreu-D…

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

I consider the geometry of the general class of scalar 2nd-order differential equations with parabolic symbol, including non-linear and non-evolutionary parabolic equations. After defining the appropriate $G$-structure to model parabolic equations, I apply Cartan techniques to determine local geometric invariants (quan…

I consider the existence and structure of conservation laws for the general class of evolutionary scalar second-order differential equations with parabolic symbol. First I calculate the linearized characteristic cohomology for such equations. This provides an auxiliary differential equation satisfied by the conservatio…

The paper studies frequency monotonicity for solutions of nonlinear equations under Ricci flow.

Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.

Sharp estimates for parabolic equations on manifolds using symmetrization.

A notion of parabolic C-subsolutions is introduced for parabolic equations, extending the theory of C-subsolutions recently developed by B. Guan and more specifically G. Székelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows.

Improved convergence and curvature estimate for parabolic Allen-Cahn equation.

Maximal regularity for nonuniformly parabolic problems with normal degeneration.

The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.

Proves smooth solution uniqueness and long-term existence for a parabolic equation on a complex manifold.

New method for analyzing elliptic and parabolic equations.

Unified diffusive bounds for non-linear parabolic equations.

Paper establishes $L^{\infty}$ estimates for complex Monge-Ampere and Hessian equations.

Develops a new parabolic equation for surfaces, proving long-time existence and convergence.

We observe that the comparison result of Barles-Biton-Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow.

Alternative proof of a theorem using parabolic Monge-Ampère equation in HKT geometry.

Paper defines parabolic frequency for Ricci flow solutions, proving monotonicity and uniqueness.

Researchers solve a nonlocal parabolic equation on manifolds using source-to-solution maps.

Extends parabolic study to flat hyperkähler manifolds.

We survey recent work on local well-posedness results for parabolic equations and systems with rough initial data.

We prove $C^\infty$ convergence for suitably normalized solutions of the parabolic complex Monge-Ampère equation on compact Hermitian manifolds. This provides a parabolic proof of a recent result of Tosatti and Weinkove.

The paper derives new gradient and Hessian estimates for nonlinear parabolic equations.

The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.

We find normal forms for parabolic Monge-Ampere equations. Of these, the most general one holds for any equation admitting a complete integral. Moreover, we explicitly give the determining equation for such integrals; restricted to the analytic case, this equation is shown to have solutions. The other normal forms exha…

Study proves long-term solutions to a specific equation on hyperKähler manifolds.

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

This survey paper is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models …

Established concavity principle for curved spaces.

Paper solves a complex equation for smooth domains.

Study shows long-term solutions for complex equations on curved spaces.

Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.