arXiv research

Study on special Lagrangian curvature potential equation, proving existence and uniqueness of smooth solutions.

The paper derives estimates for linear potentials and applies them to improve Hausdorff dimensions of singular sets in conformal geometry.

The paper classifies solutions to semilinear equations on curved spaces.

This work extends elasticity theory to curved spaces, solving stress potentials.

Study on consensus formation in manifolds with curvature constraints.

If the potential vector field of an $η$-Ricci soliton is of gradient type, using Bochner formula, we derive from the soliton equation a Laplacian equation satisfied by the potential function $f$. In a particular case of irrotational potential vector field we prove that the soliton is completely determined by $f$. We gi…

New conformal geometry method solves Einstein-Weyl equations.

Paper shows how solutions to Allen-Cahn converge to multiphase mean curvature flow.

We extend the Mason-Newman Lax pair for the elliptic complex Monge-Ampère equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. We identify the real and imaginary parts of the potential, which we call partner symmetries, w…

Prescribing $σ_k$ curvature equations are fully nonlinear generalizations of the prescribing Gaussian or scalar curvature equations. Given a positive function $K$ to be prescribed on the 4-dimensional round sphere. We obtain asymptotic profile analysis for potentially blowing up solutions to the $σ_2$ curvature equatio…

The paper introduces a new system of equations for Hessian-cscK metrics.

We consider, in the Euclidean setting, a conformal Yamabe-type equation related to a potential generalization of the classical constant scalar curvature problem and which naturally arises in the study of Ricci solitons structures. We prove existence and nonexistence results, focusing on the radial case, under some gene…

Solutions to scalar curvature equations have the property that all possible blow-up points are isolated, at least in low dimensions. This property is commonly used as the first step in the proofs of compactness. We show that this result becomes false for some arbitrarily small, smooth perturbations of the potential.

The paper solves pseudo-convexity for special Lagrangian equations, with applications in mirror symmetry.

We study the Calabi-Yau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function. Under a positive curvature condition, we show that the conjecture holds.

It is proved that solutions of the complex Monge-Ampère equation on compact Kähler manifolds with right hand side in $L^p, p>1$ are uniformly Hölder continuous under the assumption on non-negative orthogonal bisectional curvature.

Paper introduces p-Laplace equations for curvature in conformal geometry.

We develop heat kernel and Green's function estimates for manifolds with positive bottom spectrum. The results are then used to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds with Ricci curvature bounded below. As an application, we show that the curvature of a steady …

In this note we discuss how several results characterizing the qualitative behavior of solutions to the nonlinear Poisson equation can be generalized to harmonic maps with potential between complete Riemannian manifolds. This includes gradient estimates, monotonicity formulas and Liouville theorems under curvature and …

We consider a natural mechanical system on a Finsler manifold and study its \emph{curvature} using the intrinsic Jacobi equations (called \emph{Jacobi curves}) along the extremals of the least action of the system. The curvature for such a system is expressed in terms of the Riemann curvature and the Chern curvature (i…

Classifies regularity for Lagrangian mean curvature type equations.

Analyticity of heat equation extended to Bakry-Émery Ricci curvature manifolds.

The paper studies dual catenaries in the dual plane, deriving equations and characterizations.

Researchers solve porous medium equation on noncompact manifolds with Ricci curvature.

In this paper, we generalize our apriori estimates on cscK(constant scalar curvature Kähler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture…

We prove certain localized and global differential Harnack inequality for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "$…

The paper characterizes potential functions whose level sets are orbits in mechanical systems.

We establish a simple relation between curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regard…

In this paper we consider an Einstein-type equation which generalizes important geometric equations, like static and critical point equations. We prove that a complete Einstein-type manifold with fourth-order divergence-free Weyl tensor and zero radial Weyl curvature is locally a warped product with $(n-1)$-dimensional…

We establish a point-wise gradient estimate for $all$ positive solutions of the conjugate heat equation. This contrasts to Perelman's point-wise gradient estimate which works mainly for the fundamental solution rather than all solutions. Like Perelman's estimate, the most general form of our gradient estimate does not …

5D shrinking Ricci solitons with constant scalar curvature are rigid.

Geometrical aspects of a perfect fluid spacetime are described in terms of different curvature tensors and $η$-Ricci and $η$-Einstein solitons in a perfect fluid spacetime are determined. Conditions for the Ricci soliton to be steady, expanding or shrinking are also given. In a particular case when the potential vector…

We introduce a variation of the classical Ricci flow equation that modifies the unit volume constraint of that equation to a scalar curvature constraint. The resulting equations are named the Conformal Ricci Flow Equations because of the role that conformal geometry plays in constraining the scalar curvature. These equ…

Develops potential theory for WZW equation in Kähler potentials space.

We prove that every entire self-shrinking solution on $\mathbb{C}^n$ to the Kähler-Ricci flow with strictly real convex potential must be quadratic. The very same argument also gives a pointwise proof for the rigidity of entire self-shrinking solutions to Lagrangian mean curvature flow in pseudo-Euclidean space obtaine…

We study the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new conformal invariant, which is sensitive to the presence of the initial data for …

Study on line bundle flow on Kähler surfaces converging to a singular solution.

Study of a generalized Konno--Oono system with conservation laws and surface immersions.

In this paper a convergent series expansion is constructed to solve the prescribed mean curvature equation for n-dimensional hypersurfaces in n+1 dimensional Euclidean or Minkowskian space(time) which are graphs of a smooth real function u, and whose mean curvature function H is not too large in Hoelder norm, and integ…

Article provides Bernstein gradient estimates for heat equations with potential terms.

Study magnetic geodesics on Kähler potentials using variational methods.

Study on ground states of semilinear elliptic equations with various potential wells.

We apply conformal flows of metrics restricted to the orthogonal distribution $D$ of a foliation to study the question: Which foliations admit a metric such that the leaves are totally geodesic and the mixed scalar curvature is positive? Our evolution operator includes the integrability tensor of $D$, and for the case …

We study the relation between the centro-affine geometry of star-shaped planar curves and the projective geometry of parametrized maps into $\RP^1$. We show that projectivization induces a map between differential invariants and a bi-Poisson map between Hamiltonian structures. We also show that a Hamiltonian evolution …

The paper connects Schrödinger equations to geodesics on a 2-surface.

The study explores rigid constraints on almost Ricci-Bourguignon solitons on contact metric three-manifolds.

The paper introduces new equations in Kähler geometry and proves their solutions and convexity.

Researchers derive the chemical potential equation for ideal agent systems.