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.
The study proves that certain hypersurfaces in elliptic space forms are topologically rigid.
problem Characterizing the topological structure of hypersurfaces in elliptic space forms.
method Proving topological rigidity under curvature constraints and using the Gauss map.
result Hypersurfaces in elliptic space forms are diffeomorphic to spheres or their quotients.
We compute the Chern-Simons transgressed forms of some modularly invariant characteristic forms, which are related to the elliptic genera. We study the modularity properties of these secondary characteristic forms and the relations among them. We also compute the Chern-Simons forms of some vector bundles over free loop…
The paper provides estimates for eigenvalues of elliptic differential problems.
problem Computing eigenvalue estimates for elliptic differential problems.
method Analytical computation of eigenvalues for specific types of elliptic differential equations.
result Universal estimates of eigenvalues and gaps between consecutive eigenvalues are derived.
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.
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.
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.
The paper finds inequalities for eigenvalues of fourth order elliptic operators on Riemannian manifolds.
problem Eigenvalue inequalities for fourth order elliptic operators on Riemannian manifolds.
method Analyzes eigenvalues of fourth order elliptic operators in divergence form with Dirichlet boundary conditions on bounded domains in compact Riemannian manifolds.
result General inequalities for eigenvalues are derived.
Proves existence of elliptic Reeb orbit on real projective 3-space using ECH.
problem Existence of elliptic Reeb orbits on real projective 3-space.
method Use of ECH (Embedded Contact Homology) to find distinguished pseudoholomorphic curves.
result Existence of elliptic Reeb orbit proven for some contact forms on RP3. It is an interesting question whether a given equation of motion has a periodic solution or not, and in the positive case to describe them. We investigate periodic magnetic curves in elliptic Sasakian space forms and we obtain a quantization principle for periodic magnetic flowlines on Berger spheres. We give a criteri…
The paper estimates eigenvalues for specific differential operators on curved spaces.
problem Estimating eigenvalues for a class of elliptic differential operators on Riemannian manifolds.
method Analyzes eigenvalue estimates for a broader class of elliptic differential operators in divergence form.
result Provides eigenvalue estimates for Gaussian shrinking solitons and specific domains.
Quadratic points on surfaces are studied in projective 3-space.
problem Understanding quadratic points on surfaces in projective 3-space.
method Analyzing singularities of 3-webs and line fields, using Loewner's conjecture.
result Generically, the index of quadratic points is 1/3 or -1/3 for 3-webs and 1 or -1 for line fields.
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.
We derive some elliptic differential inequalities from the Weitzenböck formulas for the traceless Ricci tensor of a Kähler manifold with constant scalar curvature and the Bochner tensor of a Kähler-Einstein manifold respectively. Using elliptic estimates and maximum principle, some Lp and L∞ pinching result…
The abstract conjectures a formula for virtual elliptic genera of sheaves on surfaces.
problem Computing virtual elliptic genera of sheaves on surfaces.
method Expressed conjectures in terms of Igusa cusp form, quasi-Jacobi forms, and universal functions.
result Conjectures a formula involving virtual cobordism classes and universal functions.
Study Liouville action on quasi-Fuchsian groups, proving formulas and relating to holography.
problem Analyzing Liouville action for quasi-Fuchsian groups with different types of elements.
method Derived formulas for classical Liouville action, proved first and second variations, and established holography principle.
result Established an equality linking Liouville action and renormalized volume for quasi-Fuchsian groups.
Study special Lagrangian submanifolds with edge singularities using elliptic theory.
problem Characterize the moduli space of deformations of special Lagrangian submanifolds with edge singularities.
method Use elliptic theory for edge-degenerate differential operators on singular manifolds.
result Obtain a general theorem describing the local structure of the moduli space.
Constructs maps from field theories to complexified K-theory and elliptic cohomology.
problem Identifying geometric models for Chern characters in supersymmetric field theories.
method Higher-dimensional generalization of Fei Han's method, involving super moduli spaces and derived geometry.
result Provides evidence for the Stolz--Teichner program and geometric models for Chern characters.
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…
Sharp inequalities for submanifolds in space forms derived from elliptic operators.
problem Optimal upper bound for eigenvalues of elliptic operators on submanifolds.
method Using a general symmetric, positive definite tensor T, derived an upper bound for the second eigenvalue of elliptic operators.
result Optimal upper bound for eigenvalues given in terms of integration involving tensor properties and normal vector field.
The study finds unique and non-trivial surfaces in complex spaces.
problem Determining complete surfaces with parallel mean curvature.
method Explicit determination of surfaces in complex projective and hyperbolic planes.
result Existence and uniqueness of surfaces in positive curvature, and non-trivial surfaces in negative curvature.
The paper extends elliptic genus and proves anomaly cancellation formulas for almost complex manifolds.
problem Anomaly cancellation formulas for almost complex manifolds.
method Extended elliptic genus, proved weak Jacobi forms, derived SL_2(Z) modular forms.
result New anomaly cancellation formulas of characteristic forms for almost complex manifolds.
Proves regularity of geodesic equation on Hermitian manifolds.
problem Regularity of geodesic equation in mixed volume forms space.
method Ellipticity conditions, uniform Laplacian estimates, explicit subsolutions.
result Existence of unique C1,1 solution to Donaldson equation. New elliptic genera defined for spin manifolds.
problem Defining new elliptic genera for spin manifolds.
method Defined new two-variable elliptic genera using Jacobi forms and modular forms.
result Anomaly cancellation formulas derived from modular forms.
Defines new two-variable elliptic genera for manifolds and derives modular forms.
problem Develops new elliptic genera for manifolds.
method Introduces and defines new two-variable elliptic genera for manifolds and derives their properties.
result Derives modular forms from the elliptic genera.
Special Liouville metrics with Ricci-like conditions are determined by elliptic functions.
problem Characterizing Liouville metrics with Ricci-like conditions in complex space forms.
method Analyzing necessary conditions for induced metrics of parallel mean curvature surfaces and proving the existence of specific Liouville metrics.
result Explicit determination of special Liouville metrics with Ricci-like conditions by elliptic functions.
S. Donaldson introduced a metric on the space of volume forms, with fixed total volume on any compact Riemmanian manifold. With this metric, the space of volume forms formally has non-positive curvature. The geodesic equation is a fully nonlinear degenerate elliptic equation. We solve the geodesic equation and its pert…
Estimates gaps between eigenvalues for elliptic operators on manifolds.
problem Estimating the gaps between consecutive eigenvalues for elliptic differential operators.
method Analyzes a class of second-order elliptic differential operators in divergence form with Dirichlet boundary conditions.
result Estimates for the upper bound of gaps between eigenvalues, with results matching known best estimates for specific cases.
The paper defines new modular forms from almost complex manifolds and derives anomaly cancellation formulas.
problem Anomaly cancellation in almost complex manifolds.
method Defined a generalized elliptic genus and derived SL(2,Z) modular forms.
result Derived anomaly cancellation formulas and divisibility results for holomorphic Euler characteristic.
The generic singularities and bifurcations are classified for one-parameter families of curves with frames in a space form, the Euclidean space, the elliptic space or the hyperbolic space via projective geometry. Two kinds of frames are considered, adapted frames and osculating frames, in terms of certain differential …
We discuss critical elliptic systems in potential form. We prove existence, multiplicity, and compactness of solutions.
This is the first of a series of articles in which we are going to study the regularized determinants of the Laplacians of Calabi Yau metrics acting on (0,q) forms on the moduli space of CY manifolds with a fixed polarization. It is well known that in case of the elliptic curves the Kronecker limit formula gives an exp…
We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the Lr operator associated to immersed hypersurfaces with locally bounded (r+1)-th mean curvature Hr+1 of the space forms …
Study geometric operators on Tian-Yau spaces, finding L2 harmonic forms and asymptotic regularity.
problem Analyzing geometric elliptic operators on Tian-Yau spaces.
method Use a-pseudodifferential calculus to determine L2 harmonic forms and asymptotic regularity. result Determine the space of L2 harmonic forms and refined asymptotic regularity of ALH* structures. Estimates eigenvalues of elliptic operators with applications to mean eigenvalue bounds.
problem Estimating eigenvalues of elliptic differential operators.
method Weyl's asymptotic formula, mean eigenvalue bounds, drifting Laplacian.
result Lower bound for the mean of the first k eigenvalues of the drifting Laplacian.
The paper proposes a new model for crystallographic groups using elliptic geometry.
problem Describing the real crystallographic space using Euclidean models.
method Presented 230 crystallographic groups as elliptic motions in a closed space V3. result A special geometric model RE for crystal structures is proposed. Study finds strictly convex surfaces with specific curvature and boundary in space forms.
problem Finding strictly locally convex hypersurfaces with prescribed curvature and boundary in space forms.
method Using C2 a priori estimates and degree theory arguments, the study establishes existence results. result Existence of strictly locally convex hypersurfaces with prescribed curvature and boundary in space forms.
Study on K3 surfaces' collapsing and special Kähler structures.
problem Understanding the structure of K3 surfaces' collapsing metrics.
method Analyzing M2 and establishing connections to SKSs and Jacobian elliptic K3 surfaces. result Established a bijection between integral singular SKSs on P1 and Jacobian elliptic K3 surfaces. Local index theorem for cofinite hyperbolic Riemann surfaces derived from computational perspective.
problem Deriving the local index theorem for cofinite Riemann surfaces.
method Using Ahlfors' variational formulas and projection formulas, deriving integral formulas for variations of determinants.
result Explicit integral formulas for variations of logdetΔn and logdetNn. Deroin and Tholozan's representations are mapped to complex projective space via action-angle coordinates.
problem Mapping representations of a punctured sphere into PSL(2,R) to a simpler geometric space. method Polygonal model and chains of triangles to extract action-angle coordinates.
result Action-angle coordinates give an explicit isomorphism and almost global Darboux coordinates.
A new method integrates forms on Riemann surfaces, leading to modular forms.
problem Integrating differential forms with poles on Riemann surfaces.
method Simple procedure to integrate differential forms with arbitrary holomorphic poles, establishing an analytic theory for integrals over configuration spaces.
result Regularized graph integrals on elliptic curves are almost-holomorphic modular forms.
Uniformly elliptic Weingarten spheres in S2xR are congruent to a canonical example.
problem Characterizing uniformly elliptic Weingarten spheres in S2xR.
method Proving bounded second fundamental form and applying Hopf's result.
result Rotational uniformly elliptic Weingarten surfaces in S2xR are congruent to the canonical example.
The study proves planes are the only complete uniformly elliptic Weingarten multigraphs.
problem Proving planes are the only complete uniformly elliptic Weingarten multigraphs.
method Proving planes are the only complete multigraphs with quasiconformal Gauss map and bounded second fundamental form.
result Proves planes are the only complete uniformly elliptic Weingarten multigraphs.
Study non-symmetric diffusions on RCD spaces, proving their convergence.
problem Analyzing non-symmetric diffusion processes on RCD spaces.
method Constructing diffusion processes with Dirichlet forms, investigating conservativeness and weak convergence.
result Established convergence of diffusion laws under geometric and coefficient convergences.
Researchers compute the cohomology of an elliptic tangent bundle.
problem Computing the cohomology of a specific Lie algebroid.
method Direct computation of cohomology.
result The cohomology of the elliptic tangent bundle is computed.
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.
New non-trivial Kaehler-Ricci solitons found in infinite dimensional complex space forms.
problem Non-trivial Kaehler-Ricci solitons in infinite dimensional complex space forms.
method Exhibited families of non trivial radial Kaehler-Ricci solitons in infinite dimensional complex space forms.
result Non-trivial Kaehler-Ricci solitons exist in infinite dimensional complex space forms, contradicting previous finite dimensional results.
We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous s…