Study index theory for foliated manifolds with boundary using blup groupoids.
problem Index theory for foliated manifolds with boundary.
method Use blup groupoids and pseudodifferential calculus to study index problems.
result Construct and prove new index theorems for fully elliptic operators.
Solves nonlinear problems on metric structures through eigenvalue counting.
problem Nonlinear equations on metric structures
method Counting large eigenvalues of linearized operators
result Solves fully nonlinear Loewner-Nirenberg and Yamabe problems
Proves Lipschitz continuity for solutions of certain nonlinear elliptic equations.
problem Interior Lipschitz regularity for continuous viscosity solutions of nonlinear degenerate elliptic equations.
method Establishes interior Lipschitz regularity using a weak form of the strong comparison principle.
result Weak form of the strong comparison principle (principle of propagation of touching points) for specific operators.
Proof that certain complex equations have only simple solutions.
problem Characterizing solutions to complex equations.
method Analyzing fully nonlinear elliptic operators.
result Entire smooth solutions are quadratic polynomials.
Solves curvature problems on manifolds with negative curvature.
problem Prescribed curvature problems on closed manifolds with negative curvature.
method Investigates fully nonlinear prescribed curvature problems for modified Schouten tensor on closed Riemannian manifolds with negative curvature.
result Proves solvability of curvature problems under certain conditions.
Geometric approach simplifies K-homology computation for Lie manifolds.
problem Computing the Fredholm index of fully elliptic operators on Lie manifolds.
method Adapting geometric K-homology concepts, introducing geometric cycles and a comparison map.
result Reduction of index computation to Dirac operator index with a smoothing operator.
Derives numerical formulas for elliptic differential operators on specific groupoids.
problem Index problem of elliptic differential operators on boundary groupoids.
method Similar to Moroianu and Nistor's renormalized trace approach, focusing on eta and Atiyah-Singer terms.
result For q≥3, K-theoretic and Fredholm indices are given by the Atiyah-Singer term. Analyzes solutions to non-elliptic equations on bounded domains.
problem Analyzes solutions to non-elliptic equations on bounded domains.
method Analyzes solutions to non-elliptic equations on bounded domains.
result Analyzes solutions to non-elliptic equations on bounded domains.
In this short note, we solve a Dirichlet problem for a fully nonlinear elliptic equation. The operator is introduced by S. Donaldson and it is relevant to the geometry of the space of volume forms.
Study on maximum principles for nonlinear equations on Riemannian manifolds.
problem Investigating strong maximum principles for fully nonlinear equations on Riemannian manifolds.
method Analyzing scaling conditions and applying to various nonlinear operators.
result Established new strong comparison principles for second order uniformly elliptic problems.
Interdisciplinary study linking potential theory and elliptic PDEs.
problem Understanding solutions to nonlinear elliptic PDEs.
method Combining geometric and potential theory approaches.
result Validity of comparison principle and existence/uniqueness of solutions.
Proves solutions to elliptic equations on Hermitian manifolds with optimal conditions.
problem Solving elliptic equations on Hermitian manifolds with boundary conditions.
method Derives quantitative boundary estimates and proves existence of solutions.
result Proves existence of solutions under almost optimal structural conditions.
Proves existence and compactness of solutions to σ2-Nirenberg problem on sphere.
problem Existence and compactness of solutions to σ2-Nirenberg problem on S2. method Establishes Liouville type theorems, a priori estimates, and uses degree theory.
result Proves existence of at most one blow-up point for solutions to σ2-Nirenberg problem. Study fully nonlinear equations on Hermitian manifolds to find metrics with specific curvature.
problem Finding conformal Hermitian metrics with prescribed curvature functions.
method Blow-up argument and partial uniform ellipticity.
result Our assumptions are almost sharp, with some geometric function theory obstructions.
Estimates for polynomial operators using determinant majorization and subharmonics.
problem Bounding solutions of polynomial operators on Euclidean domains.
method Combines Alexandrov estimate and determinant majorization, using subharmonics and semiconvex approximation.
result Includes classical Alexandrov-Bakelman-Pucci estimate for linear operators.
Study fully nonlinear elliptic equations on complex manifolds.
problem Solving fully nonlinear elliptic equations on complex manifolds.
method Derive C2,α-estimate and prove existence theorems for solutions and Dirichlet problems. result Existence theorems for solutions on closed Hermitian manifolds with unbounded conditions.
We prove the existence of non-smooth solutions to fully nonlinear uniformly elliptic equations.
Uniform estimates for complex equations on compact manifolds found.
problem Uniform estimates for (n−1)−form fully nonlinear PDEs on compact Hermitian manifolds. method Local comparison with Monge-Ampère equations and finding an appropriate elliptic operator.
result A priori L∞ estimate for the equations. Study improves understanding of solutions to complex equations in geometry.
problem Understanding solutions to fully nonlinear elliptic equations.
method Obtained local second derivative estimates for strong solutions.
result Improved estimates for W2,p-strong solutions. Solves open problems for fully nonlinear elliptic equations on manifolds.
problem Solving fully nonlinear elliptic equations on manifolds.
method Analytic slope invariant and Nakai-Moishezon criterion.
result Solves open problems including hessian and hessian quotient equations.
We prove that there is no nontrivial homogeneous order 2 solutions of fully nonlinear uniformly elliptic equations in dimension 4.
Estimates derived for solutions of Neumann problems on Riemannian manifolds.
problem Gradient and second order estimates for solutions of fully nonlinear elliptic equations on compact Riemannian manifolds.
method Derivation of gradient and second order {\em a priori} estimates.
result Existence and regularity results for solutions of Neumann problems.
One way to geometrically encode the singularities of a stratified pseudomanifold is to endow its interior with an iterated fibred cusp metric. For such a metric, we develop and study a pseudodifferential calculus generalizing the Φ-calculus of Mazzeo and Melrose. Our starting point is the observation, going back to Mel…
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.
This paper is the first of two papers constructing a calculus of pseudodifferential operators suitable for doing analysis on Q-rank 1 locally symmetric spaces and Riemannian manifolds generalizing these. This generalization is the interior of a manifold with boundary, where the boundary has the structure of a tower of …
Solves a specific Dirichlet problem on Riemannian manifolds.
problem Dirichlet problem for degenerate fully nonlinear elliptic equations on Riemannian manifolds.
method Derives existence of C1,1-solutions under appropriate assumptions. result Existence of C1,1-solutions. The paper establishes boundary estimates for solutions to elliptic equations on Hermitian manifolds.
problem Boundary estimates for solutions to fully non-linear elliptic equations on Hermitian manifolds.
method Unified approach using quantitative boundary estimates, gradient estimates, and existence results.
result Established gradient estimates and unified approach to Dirichlet problem solutions.
Mathai, Melrose, and Singer compute the index of projective elliptic operators.
problem Computing the index of projective elliptic operators on manifolds with Azumaya bundles.
method Equivariant index of transversally elliptic operators as pullbacks of projective elliptic operators.
result Comprehensive fractional index formula for projective elliptic operators.
We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. Under the assumption of cone condition, we derive the L∞ estimate directly.
We prove estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. These equations are not arbitrary, but arise naturally in the study of conformal geometry.
Study fully nonlinear elliptic equations on compact hyperhermitian manifolds.
problem Solving fully nonlinear elliptic equations on compact hyperhermitian manifolds.
method Adapting Székelyhidi's approach to the hypercomplex setting, proving a priori estimates.
result Proves solvability of quaternionic Hessian and Monge-Ampère equations on compact flat hyperkähler manifolds.
We solve the Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds under essentially optimal structure conditions, especially with no restrictions to the curvature of the underlying manifold and the second fundamental form of its boundary. The main result (Theorem 1.1) includes a new (and opt…
Study on solutions to complex equations, proving strong comparison and Liouville theorems.
problem Analyzing continuous viscosity solutions to fully nonlinear elliptic equations.
method Proving strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations.
result Liouville theorem for entire solutions, showing they are either constants or standard bubbles.
Derives estimates for geometric elliptic equations on complex manifolds.
problem Estimating solutions of geometric elliptic equations on complex manifolds.
method Derives a priori real Hessian estimates independent of the right-hand side.
result Establishes optimal C1,1 regularity of geometric envelopes. New method for analyzing elliptic and parabolic equations.
problem Analyzing elliptic and parabolic equations.
method Level set version of partial uniform ellipticity.
result Effective approach to investigate equations.
We use the octonion algebra to construct singular solutions of Hessian fully nonlinear uniformly elliptic equations in 21 or more dimensions. The regularity of these solutions is the least possible one. The same is proven for Isaacs equtions.
The paper proves non-existence and classification results for elliptic equations on specific subdomains.
problem Proving non-existence and classification results for elliptic equations on subdomains of the sphere.
method Proves non-existence and classification results for elliptic fully nonlinear degenerate conformal equations on subdomains of the sphere.
result Extends results of Escobar and Hang-Wang-Jimenez for specific subdomains.
This study explores the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators using KK-theory.
problem Analyzing the index theory of Heisenberg elliptic and transversally Heisenberg elliptic operators.
method Applying Kasparov's methodology and examining specific conditions using Fourier transform of the nilpotent group C∗-algebra. result Demonstrated enhanced methods for analyzing hypoellipticity and defined transversal Heisenberg ellipticity in a KK-theoretic context. We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C^{1+ε}, showing the opt…
Sharp L∞ estimates for fully non-linear elliptic equations on compact complex manifolds.
problem Sharp L∞ estimates for fully non-linear elliptic equations on compact complex manifolds.
method Comparison with auxiliary complex Monge-Ampère equations, Hölder-Young inequality, and De Giorgi iteration lemma.
result Improved L∞ estimates for fully non-linear elliptic equations on Kähler and Hermitian manifolds.
Estimates for complex equations on manifolds derived from a conjecture.
problem Estimating solutions to complex equations on Hermitian manifolds.
method Developed second order estimates for fully nonlinear elliptic equations with gradient terms.
result Derived global estimates for an equation related to Gauduchon's conjecture.
We present a method to derive local estimates for some classes of fully nonlinear elliptic equations. The advantage of our method is that we derive Hessian estimates directly from C0 estimates. Also, the method is flexible and can be applied to a large class of equations.
We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. We derive C∞ {\em a priori} estimates, and then prove the existence of admissible solutions. In the approach, a new Hermitian metic is constructed to launch the method of continuity.
Note proves index theorem for non-elliptic Heisenberg operators.
problem Proving index theorem for non-elliptic Heisenberg operators.
method Galois covering, Heisenberg elliptic differential operators, Γ-index theorem. result Example of Heisenberg operators with non-trivial Γ-index. We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.
Study the heat operator of a transversally elliptic operator on Lie groups.
problem Spectral properties and convergence of a heat operator on Lie groups.
method Review spectral properties, define character, estimate heat operator convergence.
result Estimate of fα(t) determines convergence of the character. K-homology classes linked to elliptic operators.
problem Defining K-homology classes for elliptic operators.
method Using uniform K-homology and principal symbols.
result Classes depend only on the operator's principal symbol.
Extends index theorem to uniformly elliptic operators on manifolds.
problem Generalizing index theorem to uniformly elliptic operators.
method Local index theorem on manifolds of bounded geometry.
result Validates multigraded elliptic uniform pseudodifferential operators.