Elliptic Chern characters and Atiyah-Witten formula generalized to double loop spaces.
problem Generalizing classical Atiyah-Witten formula to double loop spaces.
method Constructing elliptic Chern and Bismut-Chern characters, defining elliptic holonomy, and using equivariant twisted parallel transport.
result Established elliptic Atiyah-Witten formula on double loop space.
New proof for curved 3-cohom manifold rational ellipticity.
problem Rational ellipticity of curved manifolds with specific cohomogeneity.
method Proved rationally elliptic for cohomogeneity-three manifolds with positive curvature and no boundary quotient.
result Closed, simply connected, positively curved, cohomogeneity-three manifolds without boundary are rationally elliptic.
The paper introduces elliptic quasi-modular forms via moduli spaces.
problem Developing a theory of elliptic quasi-modular forms.
method Using moduli spaces and the Gauss-Manin connection.
result Presented a succinct theory of elliptic quasi-modular forms.
Study Kähler geometry on vector bundles over elliptic curves.
problem Characterize Kähler metrics on vector bundle total spaces.
method Analyzing function theory and Kähler geometry on vector bundles of degree zero.
result Biholomorphic total spaces correspond to isomorphic vector bundles.
Rational ellipticity proven for G-manifolds with specific quotient properties.
problem Conditions for rational ellipticity in G-manifolds. method Proving rational ellipticity based on quotient properties and manifold submetries.
result Compact, simply connected G-manifolds with certain quotient properties are rationally elliptic. Continuous family of elliptic operators' projections maintain Cauchy data spaces.
problem Maintaining Cauchy data spaces for a continuous family of elliptic operators.
method Elementary tools and classical results applied to operator graphs, Sobolev spaces, and Green's formula.
result Orthogonalized Calderón projections form a continuous family of projections.
New domains found in hyperbolic space solve a specific elliptic problem.
problem Solving an overdetermined elliptic problem in nontrivial exterior domains of hyperbolic space.
method Constructing nontrivial domains and solving the elliptic equation.
result Positive bounded solutions found in $C^{2,α}\left(Ω
ight) \cap H^1\left(Ω
ight)$.
We describe the moduli space of stable rank 2 parabolic bundles over an elliptic curve with 3 marked points.
problem Characterize the moduli space of stable rank 2 parabolic bundles over an elliptic curve with marked points.
method Explicitly describe the moduli space as a blow-up of an embedded elliptic curve in (CP1)3 and interpret it as the SU(2) character variety of the 3-punctured torus. result The moduli space Ms(X,3) can be described as a blow-up of an embedded elliptic curve in (CP1)3 and interpreted as the SU(2) character variety of the 3-punctured torus. Explains mapping properties of elliptic operators in conical spaces.
problem Understanding mapping properties of geometric elliptic operators in conical spaces.
method Develops an approach based on B.-W. Schulze's work.
result Illustrates versatility of results in Geometric Analysis.
The paper studies elliptic operators on manifolds with boundary.
problem Characterizing boundary conditions for elliptic operators on manifolds.
method Using Calderón projectors and mixed order Sobolev spaces, the paper describes the space of boundary values and characterizes Fredholm and regular realisations.
result Characterization of boundary conditions for elliptic operators leading to Fredholm and regular realisations.
Proves regularity for quasilinear elliptic equations in metric spaces.
problem Regularity of quasilinear elliptic equations in metric measure spaces.
method Galerkin's method as an alternative to difference quotients.
result Second-order and Lipschitz regularity for a wide class of elliptic equations.
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.
In this paper, we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle. Such projective elliptic genera are rational q-series that have topological definition and also have analytic interpretation via the fractional index theorem in…
Study of 17 surface behaviors and singularities for elliptic Weingarten equations.
problem Characterizing and understanding elliptic Weingarten surfaces and their singularities.
method Phase space analysis and classification of surface behaviors.
result Classification of 17 possible qualitative behaviors for rotational surfaces.
Established concavity principle for curved spaces.
problem Solving equations on curved spaces with nonnegative curvature.
method Applied concavity principle to elliptic and parabolic equations on locally symmetric spaces with nonnegative curvature.
result First general concavity principle on spaces with non-constant sectional curvature.
Uniqueness found for elliptic equations with drift on manifolds.
problem Finding unique solutions to elliptic equations with drift on manifolds.
method Investigation in weighted Lebesgue spaces, focusing on conditions for uniqueness.
result Sharp conditions on drift term for uniqueness in polynomial volume growth manifolds.
Local Lipschitz continuity of sub-elliptic harmonic maps into CAT(0) spaces proved.
problem Proving Lipschitz continuity of sub-elliptic harmonic maps between singular spaces.
method Analyzing sub-elliptic harmonic maps from the Heisenberg group into CAT(0) spaces.
result Local Lipschitz continuity established for sub-elliptic harmonic maps.
Extended elliptical slice sampling for infinite-dimensional spaces, proving reversibility.
problem Proving reversibility of elliptical slice sampling in infinite-dimensional spaces.
method Extended elliptical slice sampling to infinite-dimensional separable Hilbert spaces, providing an alternative proof of reversibility.
result The approach yields a positive semi-definite Markov operator, proving reversibility.
Formulae for Bäcklund transformations of hyperbolic and elliptic sine-Gordon/sinh-Gordon equations.
problem Finding solutions for specific types of equations.
method Providing superposition formulae for Bäcklund transformations.
result Algebraically obtain infinitely many solutions after first integration.
Study topological hyperbolicity of moduli spaces of elliptic surfaces.
problem Characterize the largeness of the topological fundamental group of complex varieties.
method Introduce topological hyperbolicity and provide supporting evidence for moduli spaces of elliptic surfaces.
result Establish a weak form of topological hyperbolicity for moduli spaces of elliptic surfaces of Kodaira dimension one.
The paper studies gravitational instantons with flat limits and finds elliptic regularity estimates.
problem Analyzing gravitational instantons with flat limits using elliptic analysis.
method Establishing elliptic regularity estimates and showing uniform constants for a family of metrics.
result The Laplacian is Fredholm and an isomorphism between specific weighted spaces.
New Kelvin transform for anisotropic elliptic problems.
problem Semilinear and quasilinear anisotropic elliptic problems.
method Introducing a new Kelvin-type transform in the anisotropic setting.
result New insights into anisotropic elliptic problems.
New topological obstructions found for elliptic and quasiregularly elliptic manifolds.
problem Connection between ellipticity and quasiregular ellipticity.
method Proved new topological obstructions.
result Closed manifolds of both types must have virtually abelian fundamental group.
We establish some characterizations of elliptic hyperboloids (resp., ellipsoids) in the (n+1)-dimensional Euclidean space En+1, using the n-dimensional area of the sections cut off by hyperplanes and the (n+1)-dimensional volume of regions between parallel hyperplanes. We also give a few characterizat…
Characterizes stably elliptic elements in Lie groups and their properties.
problem Understanding stably elliptic elements in Lie groups and their geometric and algebraic properties.
method Characterization through fixed point algebra and Weyl group action; relates to maximal invariant cones and compactness of order intervals.
result Connected components of stably elliptic elements can be described using Weyl group action on a compactly embedded Cartan subalgebra.
Complexity results for recognizing elliptic 3-manifolds.
problem Deciding if a 3-manifold is elliptic and its homeomorphism type.
method Using triangulations and barycentric subdivisions to find Heegaard solid tori.
result Conditions for a lens space to be elliptic and a method to find Heegaard solid tori.
We develop a general method of proving the ellipticity of boundary value problems for the stationary vacuum space time, by showing that the stationary vacuum field equations are elliptic subjected to a geometrically natural collection of boundary conditions in the projection formalism. Using this we prove the manifold …
In this paper we study the general affine differential geometry of surfaces in affine space A3. For a regular elliptical surface we define a moving frame of minimal order and get the complete system of differential invariants. As an application we classify regular elliptical surfaces of constant curvatures up to aff…
These informal notes are an expanded version of lectures on the moduli space of elliptic curves given at Zhejiang University in July, 2008. Their goal is to introduce and motivate basic concepts and constructions (such as orbifolds and stacks) important in the study of moduli spaces of curves and abelian varieties thro…
The article proves estimates on homotopy and cohomology dimensions in fibrations.
problem Estimates on dimensions of homotopy and cohomology groups in fibrations.
method Proves estimates in formal elliptic spaces and specific cases.
result Proves estimates on dimensions of homotopy and cohomology groups in fibrations.
Researchers classify cmc surfaces using Jacobi elliptic functions.
problem Classifying rotational cmc surfaces in non-Euclidean space forms.
method Lie sphere geometric description of rotational linear Weingarten surfaces.
result Explicit parametrizations of cmc surfaces in hyperbolic space.
We show that an Osserman-type inequality holds for spacelike surfaces of constant mean curvature (CMC) 1 with singularities and with elliptic ends in de Sitter 3-space. An immersed end of a CMC 1 surface is an ``elliptic end'' if the monodromy representation at the end is diagonalizable with eigenvalues in the unit cir…
In the present paper we consider a special class of spacelike surfaces in the Minkowski 4-space which are one-parameter systems of meridians of the rotational hypersurface with timelike or spacelike axis. They are called meridian surfaces of elliptic or hyperbolic type, respectively. We study these surfaces with respec…
In this paper we give a geometrical interpretation of all the second elliptic integrable systems associated to 4-symmetric spaces. We first show that a 4-symmetric space G/G0 can be embedded into the twistor space of the corresponding symmetric space G/H. Then we prove that the second elliptic system is equivalent…
We give a review of the systematic construction of hierarchies of soliton flows and integrable elliptic equations associated to a complex semi-simple Lie algebra and finite order automorphisms. For example, the non-linear Schrödinger equation, the n-wave equation, and the sigma-model are soliton flows; and the equation…
Study timelike meridian surfaces in Minkowski 4-space with specific properties.
problem Characterize timelike meridian surfaces with special properties in Minkowski 4-space.
method Analyze different classes of timelike meridian surfaces with constant Gauss curvature, mean curvature, and parallel normalized mean curvature vector field.
result Explicit solutions to PDEs describing timelike surfaces with parallel normalized mean curvature vector field.
Develops local elliptic regularity for geometrically-natural operators with low regularity coefficients.
problem Local elliptic regularity for operators with low regularity coefficients in Sobolev-type spaces.
method Rescaling estimates and multiplication results for function spaces.
result Unified set of interior estimates and regularity inference for operators with Sobolev-type coefficients.
Study proves uniform ellipticity implies uniform polyconvexity for anisotropic energy functionals.
problem Investigating uniform ellipticity and polyconvexity for anisotropic geometric energy functionals.
method Proves a variant of a recent result using real polyhedral chains.
result Uniform ellipticity of an anisotropic energy functional implies uniform polyconvexity of the integrand.
Study elliptic Weingarten surfaces in warped product space with specific curvature conditions.
problem Characterize elliptic Weingarten surfaces in warped product spaces with minimal type curvature conditions.
method Analyze surfaces with mean curvature and extrinsic curvature satisfying a specific relationship under radial symmetry of the warping function.
result Existence and uniqueness of rotationally-invariant elliptic Weingarten surfaces of minimal type in RimeshR. The paper provides new gradient estimates for solutions to a nonlinear elliptic equation on smooth metric measure spaces.
problem Gradient estimates for solutions to a specific nonlinear elliptic equation on smooth metric measure spaces.
method Nash-Moser iteration technique to obtain local gradient estimates.
result New local gradient estimates for positive solutions to the equation.
Proposes a new regularization technique for neural networks using elliptic operators.
problem Improving model behavior in underrepresented data regions.
method Modifies the empirical risk minimization objective to minimize an elliptic operator over the data domain.
result The proposed regularization technique anticipates error behavior outside the training set using existing elliptic operator theory.
We prove that the elliptic Harnack inequality (on a manifold, graph, or suitably regular metric measure space) is stable under bounded perturbations, as well as rough isometries.
Elliptic curves and braid groups linked through configuration spaces.
problem Understanding the relationship between elliptic curves and braid groups via configuration spaces.
method Constructing isomorphisms between configuration spaces and triples of elliptic curves, points, and holomorphic differentials.
result Unified exceptional sequences involving braid groups and automorphisms of free groups.
We study the Liouville action for quasi-Fuchsian groups with parabolic and elliptic elements. In particular, when the group is Fuchsian, the contribution of elliptic elements to the classical Liouville action is derived in terms of the Bloch-Wigner functions. We prove the first and second variation formulas for the cla…
The Heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerat…
The paper studies gradient estimates and Liouville theorems for a nonlinear elliptic equation on metric measure spaces.
problem Gradient estimates and Liouville theorems for positive solutions to a specific nonlinear elliptic equation.
method Analyzes the nonlinear elliptic equation \( \Delta_{V}u^{m} + \mu(x)u + p(x)u^{\alpha} = 0 \) on smooth metric measure spaces with bounded Bakry-Émery curvature.
result Establishes gradient estimates and related Liouville theorems and Harnack inequalities.
In this paper we consider the complex vector spaces of holomorphic cross-sections of homogeneous holomorphic vector bundles over elliptic adjoint orbits, and provide a sufficient condition for the vector spaces to be finite dimensional in view of root systems.
Paper studies viscosity solutions in unique Martinet spaces.
problem Properties of viscosity solutions in Martinet spaces.
method Established properties and proved uniqueness of solutions.
result Uniqueness of viscosity solutions in Martinet spaces.