New functions of positive type derived from link invariants on Jones-Thompson subgroups.
problem Understanding functions of positive type on Thompson groups.
method Alternative approach using partition functions and evaluations of link invariants.
result Evaluations of the Jones polynomial and other link invariants yield functions of positive type.
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.
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.
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.
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.
In this paper, we give a sharp spectral characterization of conformally compact Einstein manifolds with conformal infinity of positive Yamabe type in dimension n+1>3. More precisely, we prove that the largest real scattering pole of a conformally compact Einstein manifold (X,g) is less than $\ndemi -1$ if and only …
Study on m-positivity in Kähler manifolds with new Monge-Ampère-type equation.
problem Exploring m-positivity in Kähler manifolds and its geometric applications. method Generalizing pseudo-effective and big Bott-Chern cohomology classes, proposing a new Monge-Ampère-type equation.
result Proof of a form of uniqueness for solutions of the Monge-Ampère-type equation.
New diameter estimate on manifolds with positive Bakry-Émery Ricci tensor.
problem Estimating the diameter of manifolds with specific curvature conditions.
method Using generalized mean curvature comparison on excess function.
result Sharper diameter estimate than previous results.
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 Hessian of Busemann functions is positive definite on certain Hadamard manifolds.
problem Analyzing the Hessian of Busemann functions on Hadamard manifolds.
method Investigating the Hessian of Busemann functions on harmonic Damek-Ricci spaces and Hadamard manifolds.
result The Hessian of Busemann functions is positive definite on certain Hadamard manifolds.
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.
In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations: ut=ΔF(u), with F′(u)>0, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equ…
Study examines maximal domains of radial harmonic functions across different curvature types.
problem Understanding maximal domains of radial harmonic functions in various curvature settings.
method Analysis of harmonic spaces with positive, zero, and negative curvature.
result Characterization of maximal domains for radial harmonic functions in different curvature contexts.
Refined estimates for surfaces in curved spaces based on Willmore functional.
problem Estimating the position of surfaces in curved spaces accurately.
method Critical points of the Willmore functional, constrained area, refined geometric center of mass.
result Improved position estimates related to ambient scalar curvature.
Uniformises Kähler surfaces with positive curvature to complex plane.
problem Uniformisation of complete Kähler surfaces with positive sectional curvature.
method New approach using uniformly Lipschitz plurisubharmonic weight functions and weighted holomorphic functions.
result Proves any complete non-compact Kähler surface with positive sectional curvature is biholomorphic to C^2.
Study spherical Fourier transform on hypergeometric type harmonic manifolds.
problem Spherical Fourier transform on harmonic Hadamard manifolds.
method Representation of spherical functions by Gauss hypergeometric functions.
result Inversion formula, convolution rule, and Plancherel theorem are derived.
Study finds infinite sign-changing solutions for a specific equation on manifolds.
problem Existence of sign-changing solutions for a Yamabe-type equation on manifolds.
method Analyzes a specific Yamabe-type equation on manifolds with proper isoparametric functions and positive focal submanifolds.
result Proves the existence of infinite sign-changing solutions for the equation when 1<q<q∗. The study shows no non-constant positive f-harmonic functions on complete gradient shrinking Ricci solitons.
problem Existence of non-constant positive f-harmonic functions on gradient shrinking Ricci solitons. method Proved the non-existence of non-constant positive f-harmonic functions using Liouville-type theorems. result No non-constant positive f-harmonic functions on complete gradient shrinking Ricci solitons. The paper defines new types of positivity and proves properties of Schur forms for vector bundles.
problem Defining and characterizing new types of positivity for vector bundles.
method Introducing and characterizing two types of strongly decomposable positivity, proving properties of Schur forms.
result Schur forms of strongly decomposable positive vector bundles are positive or weakly positive, answering a question of Griffiths.
In this paper, we establish existence results for positive solutions to the Lichnerowicz equation of the following type in closed manifolds -Δu=A(x)u^{-p}-B(x)u^{q},\quad in\quad M, where p>1,q>0, and A(x)>0, B(x)≥0 are given smooth functions. Our analysis is based on the global existence of positive solution…
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.
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. The paper introduces a new energy density function and proves Liouville type theorems for various maps.
problem Proving Liouville type theorems for holomorphic, harmonic, and pluri-harmonic maps.
method Introducing a new energy density function and deriving Hessian estimates.
result No non-constant holomorphic map exists between certain Hermitian manifolds with specific curvature conditions.
Let $({\M}, g(t))$ be a Kähler Ricci flow with positive first Chern class. We prove a uniform isoperimetric inequality for all time. In the process we also prove a Cheng-Yau type log gradient bound for positive harmonic functions on $({\M}, g(t))$, and a Poincaré inequality without assuming the Ricci curvature is bound…
The paper proves positive solutions for a specific graph equation.
problem Finding positive solutions for a p-th Yamabe type equation on graphs. method Adopted a new approach to prove the existence of positive solutions.
result Proves the existence of a positive solution for the equation.
We show that the Dirichlet problem at infinity is unsolvable for the p-Laplace equation for any nonconstant continuous boundary data, for certain range of p>n, on an n-dimensional Cartan-Hadamard manifold constructed from a complete noncompact shrinking gradient Ricci soliton. Using the steady gradient Ricci soliton, w…
Gradient estimate for harmonic functions with boundary condition proved.
problem Proving gradient estimates for harmonic functions with boundary conditions.
method Using weighted f-harmonic functions and infinite dimensional Bakry-Emery Ricci tensor. result Gradient estimates for positive f-harmonic functions with Dirichlet boundary condition. The paper proves a conjecture about positivity preserving in Riemannian manifolds.
problem Proving positivity preserving for Lp functions on Riemannian manifolds. method New a-priori regularity result, Liouville type theorem, Brezis-Kato inequality.
result Proves a conjecture by M. Braverman, O. Milatovic, and M. Shubin (2002).
The paper studies constant Q-curvature metrics on manifolds.
problem Finding constant Q-curvature metrics on manifolds.
method Global bifurcation theory applied to a fourth-order ordinary differential equation.
result Multiplicity of positive solutions and constant Q-curvature metrics.
The paper examines functional properties on manifolds with very negative curvature.
problem Functional properties on manifolds with very negative curvature.
method New Hardy-type inequalities and first and second order inequalities.
result Functional properties typically hold in manifolds with polynomially growing negative curvature.
Proves positive mass theorems for specific types of curved spaces.
problem Analyzing mass in curved spaces with boundaries.
method Proves positive mass theorems for specific types of curved spaces with boundaries.
result Establishes conditions under which mass is positive in these spaces.
New representations on surfaces with positive cross ratios.
problem Understanding representations of surfaces with specific geometric properties.
method Using geodesic currents and Anosov representations, proving systolic inequalities.
result Systolic inequalities hold for all positively ratioed representations.
Study on positive solutions of Yamabe-type equation on spheres.
problem Existence and multiplicity of conformal metrics with constant scalar curvature.
method Reduction to an ODE and application of bifurcation theory.
result Existence of degenerate solutions and multiplicity results.
Optimizes hybrid dividend strategies in dual models with periodic and continuous payments.
problem Determining the best dividend strategy in a dual model with periodic and continuous payments.
method Generalizes results from a Brownian model to a dual (spectrally positive Lévy) model, using the scale function.
result The optimal strategy is of the hybrid-barrier type and can be expressed using the scale function.
This is a survey on cohomogeneity one manifolds with positive curvature. We discuss the known examples of this type and their geometry and the functions that describe the metric. We also describe the classification of cohomogeneity one manifolds that can admit a metric with positive curvature due to Grove-Wilking-Zille…
We show the existence of a smooth spherical surface minimizing the Willmore functional subject to an area constraint in a compact Riemannian three-manifold, provided the area is small enough. Moreover, we classify complete surfaces of Willmore type with positive mean curvature in Riemannian three-manifolds.
New solutions found for Yamabe problem on spheres with foliations.
problem Yamabe problem on spheres with singular Riemannian foliations.
method Variational methods, symmetries from foliations, Sobolev embedding theorem, Principle of Symmetric Criticality.
result Existence of sign-changing and positive solutions with specific symmetries.
The paper proves the existence of infinitely many nodal solutions to a Paneitz-type equation.
problem Proving the existence of nodal solutions to a specific type of partial differential equation.
method First, a C0−estimate for positive f-invariant solutions is proven. Then, the existence of mountain pass solutions with arbitrarily large energy is established. result The existence of infinitely many nodal solutions to the equation Δ2u−αΔu+βu=uq is proven. We study the four-genus of linear combinations of torus knots: aT(p,q) # -bT(p',q'). Fixing positive p, q, p', and q', our focus is on the behavior of the four-genus as a function of positive a and b. Three types of examples are presented: in the first, for all a and b the four-genus is completely determined by the Tri…
Study complex structures with perturbed differential operators to compute curvature-like operators and obtain vanishing results.
problem Analyzing complex structures with perturbed differential operators.
method Perturbing the standard differential operator to a first-order operator Dη and computing Bochner-Kodaira-Nakano-type formulae. result Obtained vanishing results for certain harmonic spaces and Dolbeault cohomology.
It is well known that isotopic metrics of positive scalar curvature are concordant. Whether or not the converse holds is an open question, at least in dimensions greater than four. We show that for a particular type of concordance, constructed using the surgery techniques of Gromov and Lawson, this converse holds in th…
The paper studies m-positive currents and line bundles on complex manifolds.
problem Understanding m-positive currents and their properties on complex manifolds. method Introducing m-plurisubharmonic functions, proving vanishing theorems, and regularisation theorems using viscosity solutions. result Global and local regularisation theorems for m-semi-positive currents. Study on evolving graphs of functions under mean curvature flow in R^n.
problem Proving long-time existence and convergence of special Lagrangian evolution equation.
method Consider the graph of a C2 function u on Rn, deform it by mean curvature flow, and analyze under 2-positivity assumption. result Proves long-time existence and convergence results under 2-positivity assumption, improving previous results.
Unified framework for analyzing gradient flows of measures with exponential decay of entropy.
problem Analyzing exponential decay of entropy functionals in gradient flows of measures.
method Characterization of global exponential decay behaviors using Hellinger-Kantorovich geometry, shape-mass decomposition, and Polyak-Łojasiewicz-type inequalities.
result Unified theoretical framework for gradient flows with complete analysis of exponential decay behaviors.
The paper proves Eells-Sampson type theorems for subelliptic harmonic maps.
problem Existence of subelliptic harmonic maps from sub-Riemannian to Riemannian manifolds.
method Investigates subelliptic harmonic map heat flow under non-positive sectional curvature.
result Proves Eells-Sampson type existence results for subelliptic harmonic maps.
The problem of existence of solution for the Heath-Jarrow-Morton equation with linear volatility and purely jump random factor is studied. Sufficient conditions for existence and non-existence of the solution in the class of bounded fields are formulated. It is shown that if the first derivative of the Levy-Khinchin ex…
Develops theory for Callias-type operators, proving index theorem and scalar curvature obstruction.
problem Obstructing positive scalar curvature on noncompact manifolds.
method Theory of Callias-type operators in C∗-algebras, index theorem. result Obstruction to positive scalar curvature on noncompact manifolds.
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.