New method shows disjoint set-theoretic subsolutions remain so for longer.
problem Ensuring disjointness of set-theoretic subsolutions under mean curvature flow.
method Set-theoretic approach to mean curvature flow on Riemannian manifolds.
result Disjoint subsolutions remain disjoint for longer periods if one is compact.
We construct solutions to the set-theoretic Yang-Baxter equation using braid group representations in free group automorphisms and their Fox differentials. The method resembles the extensions of groups and quandles.
This paper introduces a new homology theory for Yang-Baxter solutions.
problem Defining a homology theory for set-theoretic Yang-Baxter solutions.
method Introducing normalized homology theory and proving its split into parts.
result Set-theoretic Yang-Baxter homology can be split into normalized and degenerated parts.
A homology theory is developed for set-theoretic Yang-Baxter equations, and knot invariants are constructed by generalized colorings by biquandles and Yang-Baxter cocycles.
Finite time for subsolutions on Riemannian manifolds proved.
problem Finite extinction time for subsolutions of a specific equation on Riemannian manifolds.
method Proved finite extinction time using weighted Sobolev inequality and assumptions on p, q, and ρ.
result Weak subsolutions to the equation have a finite extinction time.
Proves bounded subsolution theorem for complex Monge-Ampère equation on compact Hermitian manifolds.
problem Complex Monge-Ampère equation with positive Radon measure on compact Hermitian manifolds.
method Proves bounded subsolution theorem.
result Establishes bounded subsolution theorem for complex Monge-Ampère equation.
Sharp sub-Gaussian bounds for subsolutions of Trudinger's equation on Riemannian manifolds.
problem Bounding weak subsolutions of Trudinger's equation on Riemannian manifolds.
method Proving sub-Gaussian upper bounds for weak subsolutions.
result The upper bounds are sharp for specific classes of manifolds, including \(\mathbb{R}^{n}\).
Unified theory for geometric flows on Hermitian manifolds.
problem Geometric flows on compact Hermitian manifolds.
method Introducing parabolic C-subsolutions for parabolic equations.
result Unified approach for studying geometric flows.
This paper develops a new homology theory for biquandles and discusses geometric realizations.
problem Constructing knot invariants using set-theoretic Yang-Baxter equation.
method Developed a normalized (co)homology theory for biquandles and geometrically realized them.
result Geometric realization of biquandles has finitely generated second homotopy group for finite biquandles.
Paper solves complex Monge-Ampère equation on almost Hermitian manifolds.
problem Solving Dirichlet problem for complex Monge-Ampère equation.
method Properties of subsolutions for fully nonlinear elliptic equations.
result Existence of C2-smooth strictly J-plurisubharmonic subsolution. Study shows various weak solutions to complex flows match, proving viscosity equals pluripotential.
problem Comparing weak solutions to complex Monge-Ampère flows.
method Examined various notions of weak subsolutions and showed they coincide.
result Viscosity solution equals pluripotential solution.
The paper studies properties of group relations induced by compatible coarse structures.
problem Properties of asymptotic resemblance relations on groups.
method Generalization of asymptotic dimension and introduction of set theoretic coupling.
result Groups with compatible coarse structures that admit a set theoretic coupling are asymptotic equivalent.
The paper proves argmin function is differentiable almost everywhere and applies to minimum principles.
problem Convexity and differentiability of argmin function.
method Analyzing convex function and applying semiconcave subsolutions.
result Argmin function is differentiable almost everywhere.
Uniform bounds for complex equations using Monge-Ampère method.
problem Bounding solutions to complex equations.
method Auxiliary Monge-Ampère equation method.
result Uniform bounds remain valid even as background metrics degenerate.
Solves Dirichlet problem for elliptic equations on Hermitian manifolds.
problem Solving Dirichlet problem for fully non-linear elliptic equations on Hermitian manifolds.
method Establishing a quantitative boundary estimate under a subsolution assumption.
result Derives solvability and regularity of the Dirichlet problem.
Two definitions of set-theoretic Yang-Baxter homology are shown to be equivalent.
problem Equivalence of two Yang-Baxter homology definitions.
method Comparison of algebraic and graphic homology theories.
result The graphic homology is equivalent to the algebraic one.
We address the restriction problem for viscosity subsolutions of a fully nonlinear PDE on a manifold Z. The constraints on the restrictions of smooth subsolutions to a submanifold X in Z determine a restricted subequation on X. The problem is to show that general (upper semi-continuous) subsolutions restrict to satisfy…
This paper solves Yang-Baxter cohomology for cyclic biquandles.
problem Yang-Baxter cohomology of cyclic biquandles.
method Completely determined free parts and computed torsion subgroups of homology groups.
result Upper bounds for torsion orders in higher dimensional homology groups.
New findings on convexity of special Lagrangian geodesics.
problem Convexity of special Lagrangian geodesics in space-time.
method Space-time coordinate transformation preserving Lagrangian angle, leading to C2 estimate. result Subsolutions in all branches of the degenerate special Lagrangian equation are bi-convex.
In this paper we characterize the degenerate elliptic equations F(D^2u)=0 whose viscosity subsolutions, (F(D^2u) \geq 0), satisfy the strong maximum principle. We introduce an easily computed function f(t) for t > 0, determined by F, and we show that the strong maximum principle holds depending on whether the integral …
New flow solves LYZ equation on Kähler manifolds.
problem Solving the LYZ equation on compact Kähler manifolds.
method Introduced a new flow and showed its longtime solution converges to the LYZ equation solution under certain conditions.
result The flow converges to a singular solution on compact Kähler surfaces under specific conditions.
Solves Dirichlet problem for specific PSH functions on Hermitian manifolds.
problem Solving Dirichlet problem for Monge-Ampère equation for (n−1)-PSH functions. method Deriving a quantitative boundary estimate under (n−1)-PSH subsolutions assumption. result Quantitative boundary estimate confirmed for specific manifolds.
There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation f(D2u)=0. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman …
Study C2 estimates for p-Hessian equations on closed manifolds.
problem Estimating solutions to p-Hessian equations on closed Riemannian manifolds. method Introducing pseudo-solutions to generalize C-subsolution and proving C1 and C2 estimates. result Proves C2 estimates for general p-Hessian equations on closed manifolds under sharp conditions. The paper studies limits of flows on Kähler surfaces, proving convergence to solutions of equations.
problem Analyzing limits of flows on Kähler surfaces and their convergence to solutions of equations.
method Using a property of limits of viscosity subsolutions.
result Proves convergence of flows to weak solutions of the Monge-Ampère equation.
Characterizations of entire subsolutions for the 1-harmonic equation of a constant 1−tensionfieldaregivenwithapplicationsingeometryviatransformationgrouptheory.Inparticular,weprovethateverylevelhypersurfaceofsuchasubsolutioniscalibratedandhenceisarea−minimizingover\mathbb{R}$; and every…
Paper aims to minimize ruin probability in insurance companies using Sparre Andersen model.
problem Minimizing ruin probability in insurance companies with Sparre Andersen surplus process.
method Markovization of the surplus process, investigation of value function's regularity, dynamic programming principle, and comparison of viscosity solutions.
result The value function is the unique constrained viscosity solution to the Hamilton-Jacobi-Bellman equation.
The paper proves growth estimates for subsolutions of quasilinear equations.
problem Proving integral estimates on the minimal growth of subsolutions of quasilinear equations.
method Integral estimates and structural assumptions on the equation.
result Proves growth estimates for subsolutions of quasilinear equations.
Researchers solve Dirichlet problem for complex Monge-Ampère equation on Hermitian manifolds.
problem Solving the Dirichlet problem for the complex Monge-Ampère equation on Hermitian manifolds with boundary.
method Weak quasi-plurisubharmonic solutions and optimal subsolution theorems for bounded and Hölder continuous quasi-plurisubharmonic functions.
result Proves continuity of solutions for measures well dominated by capacity, including Lp densities and moderate measures. Let (X,α) be a Kähler manifold of dimension n, and let [ω]∈H1,1(X,R). We study the problem of specifying the Lagrangian phase of ω with respect to α, which is described by the nonlinear elliptic equation \[ \sum_{i=1}^{n} \arctan(λ_i)= h(x) \] where λi are the eigenvalues of ω with respect …
Proves regularity of geodesic equation on Hermitian manifolds.
problem Regularity of geodesic equation in mixed volume forms space.
method Ellipticity conditions, uniform Laplacian estimates, explicit subsolutions.
result Existence of unique C1,1 solution to Donaldson equation. Sharp upper bounds found for solutions of a specific equation on Riemannian manifolds.
problem Finding upper bounds for solutions of a specific equation on Riemannian manifolds.
method Proved sharp upper estimates of weak subsolutions to the Leibenson equation on Riemannian manifolds with non-negative Ricci curvature.
result Improved and proved a conjecture about upper bounds for solutions of the Leibenson equation.
Solves Dirichlet problem for fully nonlinear equations on Hermitian manifolds.
problem Solving Dirichlet problem for fully nonlinear equations on Hermitian manifolds.
method Derived C2 estimates and gradient estimates for solutions. result Solved Dirichlet problem with admissible subsolutions in some cases.
Study complex Monge-Ampère flows on Kähler manifolds using Perron method.
problem Complex Monge-Ampère flows in big cohomology classes.
method Perron method for pluripotential subsolutions.
result Upper envelope of subsolutions is a unique pluripotential solution with regularity.
Model-theoretic aspects of exotic smoothness were studied long ago uncovering unexpected relations to noncommutative spaces and quantum theory. Some of these relations were worked out in detail in later work. An important point in the argumentation was the forcing construction of Cohen but without a direct application …
The paper proves estimates for solutions to nonlinear equations on manifolds with boundary.
problem Boundary estimates for fully nonlinear Yamabe equations on Riemannian manifolds.
method Deriving a priori second derivative estimates for subsolutions.
result Existence of smooth solutions with uniform estimates.
We define a knot/link invariant using set theoretical solutions (X,σ) of the Yang-Baxter equation and non commutative 2-cocycles. We also define, for a given (X,σ), a universal group Unc(X) governing all 2-cocycles in X, and we exhibit examples of computations.
We study a fully nonlinear equation of complex Monge-Ampere type on Hermitian manifolds. We establish the a priori estimates for solutions of the equation up to the second order derivatives with the help of a subsolution.
Proves solvability of general inverse σ_k equations with constant coefficients.
problem Solvability of general inverse σ_k equations with constant coefficients.
method Proves existence of unique solution if a C-subsolution exists.
result Confirms analytical conjecture for deformed Hermitian--Yang--Mills equation.
We prove that every acyclic normal one-dimensional real Ambrosio-Kirchheim current in a Polish (i.e. complete separable metric) space can be decomposed in curves, thus generalizing the analogous classical result proven by S. Smirnov in Euclidean space setting. The same assertion is true for every complete metric space …
A construction of Kaehler-Einstein metrics using Galois coverings, studied by Arezzo-Ghigi-Pirola, is generalized to orbifolds. By applying it to certain orbifold covers of P^n which are trivial set theoretically, one obtains new Einstein metrics on odd-dimensional spheres. The method also gives Kaehler-Einstein metric…
On a manifold with boundary, we deform the metric conformally. This induces a deformation of the Schouten tensor. We fix the metric at the boundary and realize a prescribed value for the product of the eigenvalues of the Schouten tensor in the interior, provided that there exists a subsolution.
We prove weak and strong maximum principles, including a Hopf lemma, for smooth subsolutions to equations defined by linear, second-order, partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the smooth subsolutions along this boundary…
We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in $\bC^n$. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian manifolds with boundary that equipped locally conformal Kähler metrics, provided a …
Proves existence of convex surfaces with specific curvature in hyperbolic space.
problem Existence of smooth convex surfaces with prescribed curvature and boundary.
method Proves existence under strictly locally convex subsolution assumption.
result Smooth complete strictly locally convex hypersurface existence proved.
We derive a priori C2 estimates for a class of complex Monge-Ampere type equations on Hermitian manifolds. As an application we solve the Dirichlet problem for these equations under the assumption of existence of a subsolution; the existence result, as well as the second order boundary estimates, is new even for bou…
Geodesic rays of class C^{1,1} are constructed for any test configuration of a positive line bundle L on X using resolution of singularities. The construction reduces to finding a subsolution of the corresponding Monge-Ampere equation. Geometrically, this is accomplished by the use a positive line bundle on the resolut…
The paper studies equations on almost Hermitian manifolds with estimates and existence results.
problem Solving Monge-Ampère type equations on compact almost Hermitian manifolds.
method Derives C∞ a priori estimates and obtains existence results under admissible conditions. result Existence of solutions under admissible conditions for Monge-Ampère type equations.