Smooth even solutions found for a generalized convex geometry problem.
problem Dual Orlicz-Minkowski problem in convex geometry.
method Geometric flow involving Gauss curvature and normal vectors.
result Existence of smooth even solutions for smooth even measures.
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∈[0,∞) with unif…
Proves smooth solution uniqueness and long-term existence for a parabolic equation on a complex manifold.
problem Existence and uniqueness of solutions to a parabolic equation on compact complex manifolds.
method Uses parabolic Donaldson's equation to prove existence and uniqueness of smooth solutions.
result Smooth solutions to the parabolic Donaldson's equation on compact complex manifolds exist and are unique for all time.
Smooth solutions found for a specific type of Yamabe problem.
problem Regularity of viscosity solutions to the σk-Yamabe problem in the negative cone. method Analysis of Lipschitz viscosity solutions with specific assumptions.
result Existence and smoothness of solutions away from a negligible set.
We prove that the space of smooth initial data and the set of smooth solutions of the Liouville equation are homeomorphic.
Paper finds infinitely many smooth solutions to complex geometry problem.
problem Finding smooth solutions to the Strominger system on specific geometric manifolds.
method Constructing solutions on generalized Calabi-Gray manifolds.
result Explicit construction of infinitely many solutions.
Establishes unique weak solution to complex flow on smooth manifolds.
problem Existence of weak solutions to complex flow equations.
method Applying viscosity theory to prove existence of unique solution.
result Unique global weak solution exists for smooth manifolds.
Paper finds smooth convex solutions to curvature problem.
problem Finding smooth, convex solutions to curvature problems.
method Established existence of solutions through mathematical analysis.
result Smooth, origin-symmetric, strictly convex solutions found.
Study proves smooth solutions for fractional mean curvature flow within short time.
problem Short-time existence of smooth solutions for fractional mean curvature flow.
method Established using short-time existence theorem for bounded, C^{1,1}-regular initial sets.
result Smooth solutions exist for both fractional mean curvature flow and volume preserving flow.
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…
Smooth solutions found for complex Monge-Ampère flows.
problem Regulating smoothness in complex Monge-Ampère flows.
method Analyzing initial ω-psh functions with zero Lelong number. result General Monge-Ampère flow solution is immediately smooth.
Smooth Ricci flows from metric spaces can be extended to smooth solutions.
problem Extending Ricci flows from incomplete metric spaces to complete solutions.
method Ricci-harmonic map heat flow and δ-Ricci-DeTurck flow on Euclidean balls.
result Original Ricci flow can be extended to a smooth solution on a larger ball.
Smooth solutions found for Hamiltonian stationary equations in low dimensions.
problem Finding smooth solutions to Hamiltonian stationary equations in low dimensions.
method Analyzing C1,1 solutions and deriving Ck,α estimates. result Smooth solutions exist for Hamiltonian stationary equations in dimensions n≤4. Convex solutions to a specific equation are smooth when the phase is smooth enough.
problem Regularity of solutions to the Lagrangian mean curvature equation.
method Showed regularity for convex solutions under Hölder continuity conditions on the phase.
result Convex viscosity solutions are regular if the Lagrangian phase is Hölder continuous.
Smooth flow around a helix, extending previous circle solution.
problem Constructing smooth Euler flows near helical shapes.
method Generalization of a previous circle solution approach.
result Smooth flow supported near a helix.
Donaldson flow proves unique smooth solutions on symplectic forms.
problem Existence and uniqueness of smooth solutions for symplectic forms.
method Local existence of unique smooth solutions on Besov space B21,p(M,Λ2). result Local existence of unique smooth solutions on symplectic forms.
Study on special Lagrangian curvature potential equation, proving existence and uniqueness of smooth solutions.
problem Second boundary value problem for special Lagrangian curvature potential equation.
method Method of continuity with a-priori estimate.
result Existence and uniqueness of smooth uniformly convex solutions.
Smooth solutions and classification of Dirac-Einstein equations on 3-manifolds.
problem Analyzing solutions of Dirac-Einstein equations on R3. method Proving smoothness and asymptotic behavior, classifying ground state solutions.
result Scalar part is given by Aubin-Talenti functions, spinorial part is conformal image of −21-Killing spinors on S3. Study shows solutions to certain equations form smooth manifolds.
problem Understanding moduli spaces of solutions to non-linear elliptic equations.
method Analyzes moduli spaces as derived log smooth manifolds.
result Moduli spaces of solutions are derived log smooth manifolds.
Proves smoothness and estimates for special Lagrangian solutions with semi-convexity.
problem Smoothness and estimates for special Lagrangian solutions.
method Viscosity solutions, smoothness, interior derivative estimates, sharpness of conditions.
result New Liouville theorem and effective Hessian estimates for special Lagrangian solutions.
Weak solutions found for Chern-Ricci flow on complex surfaces.
problem Existence of weak solutions for Chern-Ricci flow on compact complex surfaces.
method Through blow downs of exceptional curves and backwards smooth convergence.
result Existence of weak solutions and smoothing property proved.
Study of Nahm pole solutions over 3-manifolds, proving smoothness at boundary.
problem Analyzing Nahm pole solutions over 3-manifolds.
method Examining polyhomogeneous solutions and their expansions.
result Smoothness of sub-leading terms at boundary for Einstein 3-manifolds.
We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.
Generalizes Thurston's jiggling lemma for piecewise smooth solutions.
problem Creating piecewise smooth solutions of differential relations without homotopical assumptions.
method Jiggling arbitrary sections of E to construct solutions of R. result Generalization of Thurston's lemma for piecewise smooth solutions of differential relations.
Proves existence of smooth convex solutions to capillary curvature equations.
problem Proving existence of smooth convex solutions to capillary curvature equations.
method Gradient estimate for capillary curvature equations in half-space.
result Existence of even, smooth, strictly convex solutions for all 1<p<k+1 and θ∈(0,π/2). 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 studies how singularities evolve in inverse mean curvature flow.
problem Understanding the evolution of singularities in inverse mean curvature flow.
method Defining a weak solution, analyzing blow-up tangent cones, and proving the evolution of singularities.
result Each singularity is removed when the evolving cone becomes flat, leading to the exact waiting time for a weak solution to be smooth.
This research optimizes Andrews plots for better visual clarity in high-dimensional data.
problem Visualizing high-dimensional datasets with clarity and aesthetics.
method Developed a method to add spectral smoothing to Andrews plots to reduce visual clutter.
result Optimal spatial-spectral smoothing leads to more aesthetically pleasing and clutter-free visualizations.
Improved understanding of low-rank solutions in SDPs via smoothed analysis.
problem Finding low-rank solutions to semidefinite programs efficiently.
method Penalty function formulation and smoothed analysis to avoid worst-case matrices.
result All approximate local optima are global optima for rank-constrained SDPs under certain conditions.
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.
Proves higher regularity for anisotropic inverse mean curvature flow.
problem Higher regularity of solutions to anisotropic inverse mean curvature flow.
method Proves Harnack estimate and constructs smooth solutions from C1 initial sets. result Smooth solutions become smooth outside a compact set.
Gradient estimates for nonlinear parabolic equations on metric spaces.
problem Nonexistence of positive solutions to nonlinear parabolic equations.
method Local elliptic and parabolic gradient estimates.
result Conditions guaranteeing the nonexistence of nontrivial positive solutions.
Study smooth solutions to fractional mean curvature flow, proving uniqueness and finite extinction time.
problem Understanding evolution of surfaces with fractional mean curvature.
method Established a comparison principle and evolutions equations for fractional geometric quantities.
result Proved uniqueness and finite extinction time for compact solutions.
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 L2 smoothing estimates for the curvature tensor and use them to prove pointwise smoothing estimate…
New solutions found for Hull-Strominger system on torus bundles.
problem Finding solutions to the Hull-Strominger system with torus symmetry.
method Constructing solutions on torus bundles over K3 orbifolds.
result Smooth manifolds with complex structures have solutions to the Hull-Strominger system.
The paper constructs Ricci flow solutions for non-smooth metrics in four dimensions.
problem Constructing Ricci flow solutions for non-smooth metrics in four dimensions.
method Ricci DeTurck flow on closed manifolds with initial values in W2,2. result Constructs solutions to Ricci flow for non-smooth metrics in four dimensions.
Adapts PALM to solve NMF with smooth and sparse solutions.
problem Non-negative matrix factorization for dimensionality reduction and source separation.
method Adapted PALM for convex minimization with non-differentiable constraints.
result Solves NMF with smooth and/or sparse solutions.
Finite-time extinction and smoothing effects in fractional fast diffusion on manifolds.
problem Finite-time extinction and smoothing effects in fractional fast diffusion equations.
method Nonlinear semigroups techniques, weighted Lp spaces, fractional Green function. result Sharp extinction rates and pointwise lower bounds for solutions.
The paper proves a unique solution to a shrinking flow equation converging to a soliton.
problem Existence and uniqueness of solutions to a specific shrinking flow equation.
method Anisotropic shrinking flow with speed based on support function and curvature radii.
result Smooth convergence to a soliton for certain parameter ranges.
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…
Smooth solutions found for a curvature problem in hyperbolic space.
problem Existence of smooth complete hypersurfaces with prescribed curvature in hyperbolic space.
method Utilized Pogorelov type interior second order estimate.
result Affirmative answers for specific curvature cases in hyperbolic space.
Solves smoothing problem in Chow's connectivity theorem.
problem Smoothing problem in Chow's connectivity theorem.
method Solution provided.
result Solves the smoothing problem in Chow's theorem.
Study isotropic solutions in smooth metric measure spaces with vacuum Einstein equations.
problem Investigate isotropic solutions in smooth metric measure spaces under vacuum Einstein field equations.
method Define a weighted Einstein tensor and associated vacuum field equations. Analyze solutions for different spacetime types.
result Isotropic solutions have nilpotent Ricci operator and specific forms in 2- and 3-step nilpotent manifolds.
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…
Unique solutions found for a specific flow equation.
problem Finding unique solutions for a specific flow equation.
method Used a uniform bound for the Liouville energy and a natural space-time L2-bound for the time derivative of the solution. result Uniqueness of classical solutions for the normalised two-dimensional Hamilton-Ricci flow.
Proves existence and uniqueness of solutions to the Lp Gaussian Minkowski problem.
problem Existence and uniqueness of solutions to the Lp Gaussian Minkowski problem.
method Analyzes existence and uniqueness of solutions for different values of p.
result Existence and uniqueness of smooth solutions for p > n.
Smooth solutions found for modified mean curvature flow in Riemannian manifolds.
problem Existence of smooth solutions for modified mean curvature flow.
method A priori estimates for modified mean curvature flow in Riemannian manifolds with Killing vector field.
result Existence of smooth, entire, longtime solutions for modified mean curvature flow with smooth initial data.
Smooth solutions found for hydrodynamic equations.
problem Geodesic equations on diffeomorphism groups.
method Proving smoothness of solutions and constructing exponential maps.
result Smooth solutions match boundary conditions.