The paper calculates asymptotic expansions for specific types of oscillatory integrals.
problem Analyzing oscillatory integrals with complex phase functions.
method Using asymptotic expansions of simpler phase functions to derive results for more complex cases.
result Explicit computation of coefficients in asymptotic expansions for certain integrals.
First example of open manifold with positive Ricci curvature and non-proper Busemann function.
problem Counterexample to Busemann function properness in open manifolds with nonnegative Ricci curvature.
method Provided an open manifold with positive Ricci curvature and non-proper Busemann function.
result First example of open manifold with positive Ricci curvature and non-proper Busemann function.
The study finds unique positive harmonic functions on a ball with a specific boundary condition.
problem Uniqueness of positive harmonic functions on a unit ball with a nonlinear boundary condition.
method Analytical proof of uniqueness results.
result Proves uniqueness of positive harmonic functions on the unit ball with a nonlinear boundary condition.
The paper proves solutions for Yamabe equations on manifolds with boundary.
problem Existence and multiplicity of positive solutions for Yamabe equations.
method Use of isoparametric functions to prove existence and multiplicity results.
result Existence and multiplicity results for positive solutions of Yamabe equations.
The paper explores fibered and quasi-positive links, introducing new families and invariants.
problem Understanding the structure of L-space links and their properties. method Using the H-function as a concordance link invariant.
result Introduced a subfamily of fibered strongly quasi-positive L-space links and an infinite family of non-quasi-positive L-space links. Study adaptive sensing of Cox processes using posterior sampling and positive bases.
problem Adaptive sensing of Cox point processes with intensity function modeling.
method Model intensity function as truncated Gaussian process in positive basis, use Langevin dynamics and posterior sampling.
result Demonstrated improved sensing compared to classical Bayesian experimental design.
The paper proves positivity of a L2-torsion function for certain 3-manifolds.
problem Positivity of a L2-torsion function for 3-manifolds. method Analyzes the representation variety and uses properties of the fundamental group.
result The L2-torsion function is strictly positive for 3-manifolds with infinite fundamental group. NP-PROV separates mean and variance spaces to improve function uncertainty.
problem Neural Processes fail on out-of-domain tasks due to shared latent space uncertainty.
method Separates mean and variance into function-value-related and position-related latent spaces.
result NP-PROV achieves state-of-the-art likelihood with bounded variance in drifts.
New proof of Positive Mass Theorem using Green's function and monotonicity formula.
problem Proving the Positive Mass Theorem in Riemannian geometry.
method Established through a newly discovered monotonicity formula for Green's function.
result New proof of the Positive Mass Theorem and Riemannian Penrose Inequality.
The paper extends decay estimates to graphs with positive spectrum.
problem Proving decay estimates for nonnegative functions on graphs.
method Sharp ℓ2 decay estimates for nonnegative generalized subharmonic functions. result Extends Li and Wang's result to graphs with positive Laplacian spectrum.
Small Weyl infimum on 4-manifolds with positive scalar curvature.
problem Understanding scalar curvature on 4-manifolds.
method Analyzing the Weyl functional and comparing scalar and self-dual Weyl curvatures.
result The infimum of the Weyl functional is small on many 4-manifolds with positive scalar curvature.
The surgery technique of Gromov and Lawson may be used to construct families of positive scalar curvature metrics which are parameterised by Morse functions. This has played an important role in the study of the space of metrics of positive scalar curvature on a smooth manifold and its corresponding moduli spaces. In t…
The paper defines functions that induce bounded composition operators on RKHSs with analytic positive definite functions.
problem Characterizing functions that induce bounded composition operators on RKHSs.
method Intrinsic properties of RKHSs and asymptotic properties of orthogonal polynomials.
result Only affine transforms can induce bounded composition operators in a large class of RKHSs.
In this paper, we will give a horizontal gradient estimate of positive solutions of Δbu=−λu on complete noncompact pseudo-Hermitian manifolds. As a consequence, we recapture the Liouville theorem of positive pseudo-harmonic functions on Sasakian manifolds with nonnegative pseudo-Hermitian Ricci curvature.
Study functional confounders in causal inference, enabling estimable effects.
problem Causal inference challenges with functional confounders violating positivity.
method Functional interventions, functional positivity, gradient fields, Level-set Orthogonal Descent Estimation (LODE).
result Valid causal effect estimation under certain conditions.
New positive mass theorem for hyperbolic 3-manifolds using Green functions.
problem Proving a positive mass theorem for asymptotically hyperbolic 3-manifolds.
method Using a monotonicity formula for the Green function of the Laplace operator.
result Established a new positive mass theorem for three-dimensional manifolds.
The paper proves a new inequality for 3-manifolds with noncompact boundaries.
problem Proving positivity of a convex combination of ADM masses on 3-manifolds with noncompact boundaries.
method Obtained an integral inequality for asymptotically linear harmonic functions.
result Positivity of a convex combination of ADM masses under a positivity condition on scalar curvatures and boundary mean curvatures.
Study calculates Hessian of Busemann function on Damek-Ricci spaces.
problem Calculating Hessian of Busemann function on Damek-Ricci spaces.
method Calculates eigenvalues of Hessian and proves positive definiteness.
result Hessian of Busemann function is positive definite.
The paper classifies critical metrics on manifolds with positive isotropic curvature.
problem Classifying critical metrics on manifolds with positive isotropic curvature.
method Analyzing the volume functional and solving a differential equation.
result Critical metrics are isometric to geodesic balls in S^n or specific products when conditions are met.
This paper proves a conjecture about unique positive harmonic functions in a ball.
problem Proving the uniqueness of positive harmonic functions in a unit ball for specific parameters.
method Analyzing a partial differential equation to show the solution is constant.
result Guo-Wang's conjecture is proven for the specified parameters.
We study a functional on the boundary of a compact Riemannian 3-manifold of nonnegative scalar curvature. The functional arises as the second variation of the Wang-Yau quasi-local energy in general relativity. We prove that the functional is positive definite on large coordinate spheres, and more general on nearly roun…
New method produces positive colored superpolynomials from four-point functions.
problem Constructing positive superpolynomials for colored link invariants.
method Four-point functions from topological vertices.
result New relation between super- and hyperpolynomials.
Distance function to a finite set is a topological Morse function.
problem Characterizing the topological Morse function of a finite set.
method Analyzing the distance function to a finite set in \(\mathbb{R}^n\).
result Distance function is a topological Morse function, with precise critical points and indices.
The paper defines a new mass quantity for 3-manifolds and proves a positive mass theorem.
problem Proving the positive mass theorem for a new geometric quantity.
method Defining X-ADM mass and using a monotonicity formula. result Established a relative positive mass theorem for asymptotically flat 3-manifolds.
The paper studies Harnack inequalities on Finsler metric measure spaces.
problem Analyzing Harnack inequalities on Finsler metric measure spaces.
method Using weighted Ricci curvature and distortion conditions, the authors derive an elliptic p-Harnack inequality.
result The paper establishes an elliptic p-Harnack inequality and derives Hölder continuity and gradient estimates for positive harmonic functions.
The paper provides gradient estimates for a specific equation on Riemannian manifolds.
problem Gradient estimates for positive solutions to a specific equation on Riemannian manifolds.
method Obtained gradient bounds for positive solutions without depending on the solution's bounds or the Laplacian of the distance function.
result Gradient bound of a positive solution does not depend on the solution's bounds or the Laplacian of the distance function.
In this paper we consider Riemannian manifolds (Mn,g) of dimension n≥5, with semi-positive Q-curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a strong maximum principle; (ii) the Paneitz operator is a positive operator; and (iii) its Gree…
In this article we show that every geodesic is rank one and the Hessian of Busemann functions is positive definite for a harmonic Damek-Ricci space, a two step solvable Lie group with a left invariant metric. Moreover, the eigenspace of the Hessian of Busemann functions on a Hadamard manifold (M,g) corresponding to e…
New findings show functional inequalities fail on Finsler manifolds with positive S-curvature.
problem Failure of functional inequalities on Finsler manifolds with positive S-curvature.
method Analysis of Finsler metric measure manifolds with reversibility, flag curvature, and S-curvature.
result Functional inequalities fail on Finsler manifolds with positive S-curvature.
This paper is being replaced by another of the author's that contains a brief summary of the problem of positivity of Green's functions, heat kernels, and principal eigenvalues of higher-order elliptic differential operators.
The study examines rigidity and stability of gradient estimates on surfaces and manifolds.
problem Rigidity and stability of gradient estimates for positive harmonic functions and solutions to heat equations.
method Sharp gradient estimates for positive harmonic functions and solutions to heat equations on surfaces and manifolds with nonnegative curvature.
result Obtained rigidity and stability results for gradient estimates.
Estimates for harmonic functions in curved spaces.
problem Quantifying harmonic functions in curved spaces.
method Quantitative Sobolev estimates for p-harmonic functions in manifolds with curvature conditions. result Established a quantitative second order Sobolev estimate for p-harmonic functions. Proves Green function rigidity for specific operators and obtains new ADM mass formula.
problem Proving Green function rigidity for specific operators and obtaining new ADM mass formula.
method Positive mass theorem and positive energy theorem for Paneitz operator.
result Obtained new formula for the ADM mass of asymptotically flat hypersurfaces.
The study confirms Liouville-type theorems for positive harmonic functions on manifolds with nonnegative Ricci curvature and strictly convex boundary.
problem Proving Liouville-type theorems for positive harmonic functions on specific types of manifolds.
method Employing the P-function method and a closed conformal vector field inherent to such manifolds.
result Confirms some cases of Wang's conjecture and provides a partial verification of Wang's conjecture on warped product manifolds.
In this paper we study the asymptotic behaviour of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. This implies a full asymptotic expansion of this function on the set where the curvature of the line bundle is non-degener…
New curvature assumptions prove Nakano positivity for complex vector bundles.
problem Proving Nakano positivity for complex vector bundles under varying curvature assumptions.
method Using a variant of Hörmander's theorem, the authors show Nakano positivity under more general curvature conditions.
result Nakano positivity holds for complex vector bundles under different curvature assumptions.
The paper proves a discrete positive mass theorem for graphs.
problem Formulating and proving a discrete positive mass theorem for graphs.
method Introducing asymptotically flat graphs, defining ADM mass, and using discrete harmonic functions.
result An asymptotically flat graph with non-negative Ricci curvature is isomorphic to the standard grid graph.
Conditions for scalar curvature on compact manifolds under conformal deformation.
problem Finding conditions for scalar curvature functions on compact manifolds.
method Analyzing sufficient and necessary conditions for scalar curvature problems within conformal classes.
result Conditions for scalar curvature functions on various compact manifolds.
We show that if p:M→N is a normal Riemannian covering, with N closed, and M has exponential volume growth, then there are non-constant, positive harmonic functions on M. This was conjectured by Lyons and Sullivan in \cite{LS}.
The paper ensures positivity of solutions to stochastic equations with positive initial data.
problem Ensuring positivity of solutions to stochastic equations with positive initial data.
method Providing sufficient conditions on coefficients for positivity of mild solutions.
result Sufficient conditions for positivity of solutions to stochastic equations.
The Weyl functional is analyzed on 4-manifolds with positive scalar curvature.
problem Analyzing the Weyl functional on 4-manifolds with positive scalar curvature.
method Using integral inequalities and properties of the Weyl tensor.
result Generalized Gursky's inequality for 4-manifolds with positive scalar curvature.
The study examines exceptional sets for radial limits of superharmonic functions on curved manifolds.
problem Understanding exceptional sets for radial limits of superharmonic functions on curved manifolds.
method Analysis of radial geodesic rays, Poisson integrals, Green potentials, and Riesz decomposition.
result Sharp bounds on Hausdorff dimensions of exceptional sets for superharmonic functions.
Discuss folklore statements about manifolds with curvature bounds.
problem Distance functions in manifolds with curvature bounds.
method Regularity, subsets of positive reach, and cut locus.
result Folklore statements about manifolds with curvature bounds are discussed.
The paper proves positivity preservation and self-adjointness for Schrödinger operators on incomplete Riemannian manifolds.
problem Positivity preservation and self-adjointness for Schrödinger-type operators on incomplete Riemannian manifolds.
method Control of potential behavior near the Cauchy boundary, essential self-adjointness proof, core of smooth compactly supported functions.
result Positivity preservation and essential self-adjointness of Schrödinger operators on Lp functions on incomplete Riemannian manifolds. In our previous paper math.DG/0010008, we develop some new techniques in attacking the convergence problems for the Kähler Ricci flow. The one of main ideas is to find a set of new functionals on curvature tensors such that the Ricci flow is the gradient like flow of these functionals. We successfully find such functio…
The study proves that certain manifolds can have metrics with specific volume growth.
problem Determining if manifolds with positive scalar curvature can have metrics with a given volume growth.
method Using Gromov-Lawson and Grimaldi-Pansu constructions, the study proves the existence of metrics with the desired volume growth on specific manifolds.
result The study positively answers the question for manifolds that are infinite connected sums of closed manifolds with positive scalar curvature.
We study functions whose truncations are convex or quasiconvex.
problem Understanding functions with specific truncation properties.
method Analyzing C2-smooth functions with positive definite Hessians. result Injectivity of restricted gradient in positive definite region.
Study on a Bahri-Brezis problem on hyperbolic manifolds.
problem Existence of solutions on asymptotically hyperbolic manifolds.
method Algebraic Topological argument of Bahri-Coron.
result Existence of at least one solution under specific conditions.