arXiv research

Smooth even solutions found for a generalized convex geometry problem.

In this article, we shall investigate the relationship between the existence or non-existence of non-singular solutions to the normalized Ricci flow and smooth structures on closed 4-manifolds, where non-singular solutions to the normalized Ricci flow are solutions which exist for all time $t \in [0, \infty)$ with unif…

We prove that the space of smooth initial data and the set of smooth solutions of the Liouville equation are homeomorphic.

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

Smooth solutions found for a specific type of Yamabe problem.

Paper finds smooth convex solutions to curvature problem.

We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators $P$ can be glued in a chosen region in order to obtain a new smooth soluti…

We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positiv…

Smooth solutions found for Hamiltonian stationary equations in low dimensions.

Convex solutions to a specific equation are smooth when the phase is smooth enough.

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

Study shows solutions to certain equations form smooth manifolds.

Smooth solutions and classification of Dirac-Einstein equations on 3-manifolds.

Proves smoothness and estimates for special Lagrangian solutions with semi-convexity.

We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.

Proves existence of smooth convex solutions to capillary curvature equations.

Generalizes Thurston's jiggling lemma for piecewise smooth solutions.

In this paper we prove that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave. The value function is smooth if admissible controls satisfy an integrability condition or if it is contin…

The paper proves estimates for solutions to nonlinear equations on manifolds with boundary.

This research optimizes Andrews plots for better visual clarity in high-dimensional data.

Proves higher regularity for anisotropic inverse mean curvature flow.

We establish fundamental results for a parabolic flow of Riemannian metrics introduced by Bahuaud-Helliwell in arXiv:1010:4287v1 which is based on the Fefferman-Graham ambient obstruction tensor. First, we obtain local $L^2$ smoothing estimates for the curvature tensor and use them to prove pointwise smoothing estimate…

The paper constructs Ricci flow solutions for non-smooth metrics in four dimensions.

Finite-time extinction and smoothing effects in fractional fast diffusion on manifolds.

Solves smoothing problem in Chow's connectivity theorem.

A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors, we study the existence or non-existence of non-singular solutions of the normalized Ricci flow on 4-manifolds with non-trivial fu…

We show uniqueness of classical solutions of the normalised two-dimensional Hamilton-Ricci flow on closed, smooth manifolds for smooth data among solutions satisfying (essentially) only a uniform bound for the Liouville energy and a natural space-time $L^2$-bound for the time derivative of the solution. The result is s…

Smooth solutions found for a curvature problem in hyperbolic space.

Study isotropic solutions in smooth metric measure spaces with vacuum Einstein equations.

In this paper we investigate H-minimal graphs of lower regularity. We show that noncharactersitic C^1 H-minimal graphs whose components of the unit horizontal Gauss map are in W^{1,1} are ruled surfaces with C^2 seed curves. In a different direction, we investigate ways in which patches of C^1 H-minimal graphs can be g…

Proves existence and uniqueness of solutions to the Lp Gaussian Minkowski problem.

We prove the long time existence and uniqueness of solution to a parabolic Monge-Ampère type equation on compact Hermitian manifolds. We also show that the normalization of the solution converges to a smooth function in the smooth topology as $t$ approaches infinity which, up to scaling, is the solution to a Monge-Ampè…

Smooth solutions found for modified mean curvature flow in Riemannian manifolds.

Smooth solutions found for hydrodynamic equations.

The paper examines the smoothness of solutions to a specific partial differential equation on smooth domains.

In this paper, we study the partial convexity of smooth solutions to the heat equation on a compact or complete non-compact Riemannian manifold M or Kahler-Ricci flow. We show that under a natural assumption, a new partial convexity property for smooth solutions to the heat equation is preserved.

Modified perturbation method removes non-smoothness in solving Black-Scholes equations.

We consider a shrinking flow of smooth, closed, uniformly convex hypersurfaces in (n+1)-dimensional Euclidean space with speed fu^{alpha}{sigma}_n^{beta}, where u is the support function of the hypersurface, alpha, beta are two constants, and beta>0, sigma_n is the n-th symmetric polynomial of the principle curvature r…

We prove the existence of non-smooth solutions to fully nonlinear uniformly elliptic equations.

We prove the existence of non-smooth solutions to Special Lagrangian Equations in the non-convex case.

Study on octonionic Nahm's equations and their moduli space properties.

We apply ideas from viscosity theory to establish the existence of a unique global weak solution to the generalized Kahler-Ricci flow in the setting of commuting complex structures. Our results are restricted to the case of a smooth manifold with smooth background data. We discuss the possibility of extending these res…

We consider smooth, not necessarily complete, Ricci flows, $(M,g(t))_{t\in (0,T)}$ with ${\mathrm{Ric}}(g(t)) \geq -1$ and $| {\mathrm{Rm}} (g(t))| \leq c/t$ for all $t\in (0 ,T)$ coming out of metric spaces $(M,d_0)$ in the sense that $(M,d(g(t)), x_0) \to (M,d_0, x_0)$ as $t\searrow 0$ in the pointed Gromov-Hausdorff…

In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions equations for fractional geometric quantities that yield preservation of certain qu…

The paper provides new gradient estimates for solutions to a nonlinear elliptic equation on smooth metric measure spaces.

The study proves no smooth solutions for certain conformally invariant equations.

We construct new smooth solutions to the Hull-Strominger system, showing that the Fu-Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. In particular, we prove that, for $13 \leq k \leq 22$ and $14\leq r\leq 22$, the smooth manifolds $S^1\times \sharp_k(S^2\times S^3)$…

This paper concerns the inverse mean curvature flow of convex hypersurfaces which are Lipschitz in general. After defining a weak solution, we study the evolution of the singularity by looking at the blow-up tangent cone around each singular point. We prove the cone also evolves by the inverse mean curvature flow and e…