Derives concavity inequality and estimates for k-Hessian equations.
problem Interior estimates and curvature estimates for k-Hessian equations. method Concavity inequality and semi-convexity condition.
result Interior estimates and Liouville-type result for semi-convex solutions.
Study estimates eigenvalues for concave Hessian operators on convex domains.
problem Estimating eigenvalues for concave elliptic Hessian operators.
method Investigates Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators.
result Existence and properties of the first nonzero eigenvalue and eigenfunction.
The paper introduces new functionals and equations for complex vector bundles.
problem Complex vector bundle equations and their solutions.
method Introducing generalized Donaldson's functionals and discussing their Euler-Lagrange equations.
result Uniqueness of solutions to complex vector bundle equations.
We solve the Dirichlet problem for k-Hessian equations on compact complex manifolds with boundary, given the existence of a subsolution. Our method is based on a second order a priori estimate of the solution on the boundary with a particular gradient scale. The scale allows us to apply a blow-up argument to obtain c…
Paper solves overdetermined k-Hessian equation in exterior domains.
problem Overdetermined problem for k-Hessian equation in exterior domains. method Combining integral identities and geometric inequalities, derived general monotone formulas.
result Established general monotone formulas for k-admissible solutions. Paper shows non-convexity in solutions to Hessian equations.
problem Non-convexity of solutions to k-Hessian equations in exterior domains. method Examples and new proof for quasiconvexity of harmonic functions.
result Solutions to k-Hessian equations are not quasiconvex in exterior domains. Study proves radial symmetry of solutions to certain nonlinear equations in space forms.
problem Proving radial symmetry of solutions to nonlinear equations in space forms.
method Establishing Rellich-Pohožaev type identities for Hessian quotient and k-Hessian equations.
result Radial symmetry of solutions for Hessian quotient and k-Hessian equations in space forms.
Paper solves Minkowski problem for k-torsional rigidity.
problem Solving Minkowski problem for k-torsional rigidity.
method Constructing Hadamard variational formula, proposing k-torsional measure, using curvature flow method.
result Existence of smooth non-even solutions to the Minkowski problem.
Simplified proof for convex hypersurface curvature estimate.
problem Curvature estimate for convex hypersurfaces.
method Simple proof for Weingarten curvature equation.
result Curvature estimate for convex hypersurfaces proved.
Proves a generalized Minkowski inequality for starshaped domains.
problem Proving a generalized Minkowski inequality for smooth, (k−1)-convex starshaped domains. method Solvability of the degenerate k-Hessian equation on the exterior domain Rn∖Ω. result Generalized Minkowski inequality holds for smooth, (k−1)-convex, starshaped domains. In this paper we study integer multiplicity rectifiable currents carried by the subgradient (subdifferential) graphs of semi-convex functions on a n-dimensional convex domain, and show a weak continuity theorem with respect to pointwise convergence for such currents. As an application, the k-Hessian measures are ca…
Study Neumann problem for special Lagrangian type equations.
problem Neumann problem for special Lagrangian type equations.
method Uniform a priori estimates, continuity method, direct proof of boundary double normal derivative estimates.
result Existence result for Neumann problem of special Lagrangian type equations.
The paper introduces a new system of equations for Hessian-cscK metrics.
problem Finding constant scalar curvature Kähler metrics.
method Proposes a coupled system of complex Hessian equations and shows it can be variational.
result Proves a C0-estimate for the system that depends on entropy. Paper solves a new Minkowski problem for a specific type of rigidity.
problem Solving a new Minkowski problem for a specific type of rigidity.
method Developed a nonlinear partial differential equation and used a curvature flow method.
result Existence of smooth non-even solutions to the p-th dual Minkowski problem for p < n-2.
The paper finds convex hypersurfaces with specific curvature properties.
problem Finding convex hypersurfaces with prescribed Hessian curvatures and Gauss images.
method Used novel C2 boundary estimates based on orthogonal invariance and infinitesimal rotations. result Proved existence of strictly convex graphic hypersurfaces with prescribed k-Hessian curvatures. Solves Christoffel-Minkowski problem and Hessian equations with radial symmetry.
problem Christoffel-Minkowski problem and Hessian equations under rotational symmetries.
method Constructing explicit convex solutions to mixed Monge-Ampère equations on \(\mathbb{R}^n\) under radial symmetry.
result Explicit representation formula for the support function of the resulting convex body.
The paper proves Liouville rigidity for Hessian equations, characterizing geometric conditions for constant solutions.
problem Characterizing geometric conditions for constant solutions in Hessian equations.
method Recursive geometric condition (Liouville admissibility) and anisotropic constructions.
result The Liouville-type property is characterized as a geometric property of the admissible set.
We introduce the notion of pseudohermitian k-curvature, which is a natural extension of the Webster scalar curvature, on an orientable manifold endowed with a strictly pseudoconvex pseudohermitian structure (referred here as a CR manifold) and raise the k-Yamabe problem on a compact CR manifold. When k=1, the problem w…
This paper completes the construction of arbitrary order conformally invariant differential operators in higher spin spaces. Jan Slovák has classified all conformally invariant differential operators on locally conformally flat manifolds. We complete his results in higher spin theory by giving explicit expressions for …
The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.
problem Eigenvalue problems and heat trace asymptotics for different operators.
method Establishes connections and inequalities for eigenvalues and heat traces.
result Eigenvalue inequalities and three-term asymptotic formulas for heat traces of various operators.
Introduces a new elliptic operator with positive eigenvalue.
problem None explicitly stated in the abstract.
method Introduces a new elliptic operator called the two-radical Laplace operator.
result The eigenvalue of the new operator is the positive square root of the Laplace operator's eigenvalue.
Proves Kato inequalities for various conformal operators.
problem Proving inequalities for differential operators.
method Analyzes a class of first order differential operators, including Dirac and Penrose twistor operators.
result Derives Kato inequalities that interpolate between classical and refined versions.
We describe a set of conformally covariant boundary operators associated to the Paneitz operator, in the sense that they give rise to a conformally covariant energy functional for the Paneitz operator on a compact Riemannian manifold with boundary. These operators naturally give rise to a first- and third-order conform…
Study on biharmonic hypersurfaces with specific recurrent operators in Euclidean space.
problem Characterizing biharmonic hypersurfaces with recurrent operators.
method Analysis of various recurrent operators and their impact on biharmonic hypersurfaces.
result Some well-known recurrent operators play a significant role in making biharmonic hypersurfaces minimal.
Local index theorem for chiral geometric operators proved using heat kernel.
problem Proving a local index theorem for geometric first-order differential operators.
method Using Gilkey's invariance theory and heat kernel techniques.
result Supertrace of heat kernel converges to Chern-Weil form.
The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.
problem Spaces of unbounded Fredholm operators and their properties.
method Analyzing the spaces and proving homotopy equivalences.
result Natural maps between four spaces of unbounded Fredholm operators are homotopy equivalences.
GJMS operators connect geometry, analysis, and physics.
problem None explicitly stated; focus on operators and their impact.
method Construction of conformally invariant differential operators.
result GJMS operators have significant impact in geometry, analysis, and physics.
Extends Calabi operator to Riemannian locally symmetric spaces.
problem Local integrability conditions on Riemannian locally symmetric spaces.
method Generalizes Calabi operator to Riemannian locally symmetric spaces.
result Generalised operator works in irreducible case and fails in products.
New spectral torsion defined for rescaled Dirac operators.
problem Defining spectral torsion for rescaled Dirac operators.
method Using three vector fields and noncommutative residue.
result Computed spectral torsion for one form rescaled Dirac operators.
Proves formal self-adjointness of certain differential operators.
problem Verifying conjectures about differential operators.
method Proving formal self-adjointness through mathematical proof.
result Proves two conjectures about differential operators.
Researchers create new operators from Riemannian invariants.
problem Developing new mathematical tools for Riemannian geometry.
method Introducing formally self-adjoint conformally covariant polydifferential operators.
result Found a fourth-order, conformally covariant tridifferential operator.
The paper proves new theorems about specific types of operator perturbations.
problem Analyzing conformal perturbations of Dirac and signature operators.
method Developed Kastler-Kalau-Walze type theorems for specific operator types.
result Established new theorems for six-dimensional manifolds with boundary.
Study of Dirac-like operators on spin manifolds with large mass parameters.
problem Understanding spectra of Dirac-like operators with piecewise constant mass terms.
method Analysis of asymptotic regimes to derive effective operators.
result Extension of MIT Bag operator concept to spin geometry.
Study essential spectrum of differential operators on geometrically finite orbifolds.
problem Analyzing the essential spectrum of differential operators over specific geometric structures.
method Investigates first order and Laplace type elliptic differential operators on Riemannian vector bundles over geometrically finite orbifolds.
result Discovers properties of essential spectra for these operators.
The study proves inequalities for complex operators on curved spaces.
problem Establishing inequalities for nonlocal operators on curved spaces.
method Defining and analyzing nonlocal Pucci operators on manifolds with nonnegative sectional curvatures, proving Harnack inequalities and Holder estimates.
result Harnack inequalities and Holder estimates for nonlocal operators on manifolds with nonnegative sectional curvatures.
Study on opers over complex manifolds of dimension one.
problem Investigating opers over complex manifolds of dimension one.
method Introducing relative opers and differential operators, analyzing their equivalence.
result Bijective correspondence between relative opers and differential operators.
Mixtures of neural operators reduce active complexity in operator learning.
problem Reduction of active complexity in operator learning models.
method Constructive comparison between routed mixtures of neural operators (MoNOs) and a fixed single-neural-operator construction.
result Every scalar uniformly continuous nonlinear operator can be approximated by a MoNO whose active expert has smaller depth, width, and rank scaling.
Formula for Hadamard coefficients from Green's operators on spacetimes.
problem Calculating Hadamard coefficients from Green's operators on spacetimes.
method Developed formulas for diagonal values and integrals over the diagonal of Hadamard coefficients.
result Formulated analogues of Hadamard expansions and resolvents for Green's operators.
Study perturbs Dirac operators in any dimension, focusing on Majorana fermions.
problem Understanding perturbations of Dirac operators in various dimensions.
method Analyzes canonical perturbations of Dirac operators on Hermitian Clifford modules.
result Characterizes the low-energy spectrum of these operators on complete surfaces.
Identifies Lorentzian locally symmetric spaces where Calabi operator suffices to determine Killing operator range.
problem Determining when the Calabi operator can identify the range of the Killing operator in Lorentzian locally symmetric spaces.
method Developed criteria for a connection to be in the range of a connection, applied to the Killing connection.
result For indecomposable spaces, the Calabi operator suffices to identify the range of the Killing operator; for products, it fails.
Method calculates spectra of Rarita-Schwinger operator on symmetric spaces.
problem Calculating spectra of the Rarita-Schwinger operator on compact symmetric spaces.
method Using Weitzenböck formulas, Laplace operator, Casimir operator, Freudenthal's formula, and branching rules.
result Obtained spectra on the sphere, complex projective space, and quaternionic projective space.
Researchers create a family of conformally covariant operators.
problem Developing a comprehensive set of conformally covariant operators.
method Constructing a family of conformally covariant tridifferential operators as tangential operators in the Fefferman--Graham ambient space.
result Symmetrization of ambient operators is formally self-adjoint.
The paper revisits and analyzes the tmd-operator in almost Kähler manifolds.
problem Constructing an elliptic operator analogous to the ∂∂ operator in complex or Kähler manifolds.
method Local analysis estimates and demonstration using the Atiyah-Hitchin-Singer operator.
result Every d-exact (1,1)-form is globally tmd-exact for compact taming symplectic 4-manifolds.
Paper generalizes paracomposition and change of variables for paradifferential operators.
problem Generalizing paracomposition and change of variables for paradifferential operators in low regularity settings.
method Drops diffeomorphism hypothesis, estimates in Sobolev and Zygmund spaces, discusses pull-back of pseudodifferential and paradifferential operators.
result Sharp estimates for composition in Sobolev and Zygmund spaces, change of variables in paradifferential operators.
The k-Dirac operator is a differential operator which is natural to geometric structure of a parabolic type. We will give a set of initial conditions for this operator. In the proof of the claim we will need to adapt some parts from the theory of exterior differential systems to the setting of weighted differential ope…
We describe a set of conformally covariant boundary operators associated to the sixth-order GJMS operator on a conformally invariant class of manifolds which includes compactifications of Poincaré--Einstein manifolds. This yields a conformally covariant energy functional for the sixth-order GJMS operator on such manifo…
The paper derives expansions for Green's operators and resolvents using Hadamard methods.
problem Analyzing normally hyperbolic operators and their Green's functions.
method Hadamard expansions for powers of Green's operators and resolvents.
result Derives expansions involving Hadamard coefficients for advanced/retarded Green's operators.
The paper quantizes Kähler manifolds using differential operators.
problem Quantizing classical observables on Kähler manifolds as differential operators.
method Constructing higher-order differential operators using Fedosov-type constructions and proving asymptotic equivalence to Berezin-Toeplitz operators.
result Holomorphic differential operators are precisely those that arise as Berezin-Toeplitz operators for quantizable functions.