Study on harmonic maps on weighted Riemannian foliations.
problem Characterize harmonic maps on weighted foliations.
method Analyze transversally f-harmonic and (F,F′)f-harmonic maps. result Equivalence of transversally f-harmonic and (F,F′)f-harmonic maps in minimal foliations. We study the transversally harmonic maps between foliated Riemannian manifolds. In particular, we prove that under some curvature conditions, any transversally harmonic map is transversally totally geodesic.
Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.
problem Generalizing Schwarz lemma for harmonic maps.
method Using Bochner techniques and sub-Laplacian comparison theorem.
result Established a generalization of Schwarz lemma for transversally harmonic maps.
In the paper, we study variation formulas for transversally harmonic maps and bi-harmonic maps, respectively. We also study the transversal Jacobi field along a map and give several relations with infinitesimal automorphisms.
The paper studies critical points of horizontal energy functional in Riemannian foliations.
problem Analyzing critical points of horizontal energy functional in Riemannian foliations.
method Utilizing stress-energy tensor, establishing monotonicity formulas, and Jin-type theorems.
result Established monotonicity formulas for horizontally harmonic maps and transversally harmonic maps.
We study the Liouville type theorems for transversally harmonic and biharmonic maps on foliated Riemannian manifolds
The paper proves a Liouville theorem for a specific type of harmonic maps on foliated manifolds.
problem Investigating harmonic maps on foliated Riemannian manifolds.
method First variational formulas, generalized Weitzenböck type formula, and Liouville type theorem for (F,F′)p-harmonic maps. result Established a Liouville type theorem for (F,F′)p-harmonic maps. Study on determining metrics via Dirichlet-to-Neumann map for harmonic maps.
problem Determining Riemannian metrics from boundary measurements.
method Higher linearization method, integral identities, energy rigidity.
result Metrics on the target manifold are equal if the target is analytic.
Harmonic maps study on surfaces with non-positive curvature.
problem Characterize harmonic maps on surfaces with non-positive curvature.
method Transversality theory for Banach manifolds.
result Set of somewhere injective harmonic maps is open, dense, and connected.
Maps and measures on surfaces link best Lipschitz and least gradient functions.
problem Analyzing maps between surfaces and their geometric properties.
method Duality between best Lipschitz and least gradient maps, geodesic laminations, and transverse measures.
result The infinity harmonic map defines a geodesic lamination and the least gradient map defines a transverse measure.
Paper defines and studies measures related to earthquakes and best Lipschitz maps.
problem Understanding Thurston's conjecture about maps and measures on hyperbolic surfaces.
method Examining Lie algebra valued transverse measures and their relation to earthquakes.
result Defines and shows correspondence between best Lipschitz maps and earthquakes.
The paper explores a generalized notion of transversality in harmonic analysis.
problem Generalizing transversality to collections of submanifolds in harmonic analysis.
method Examining three related concepts of transversality in harmonic analysis and showing their equivalence.
result Three concepts of transversality in harmonic analysis are equivalent.
We study the transversal hard Lefschetz theorem on a transversely symplectic foliation. This article extends the results of transversally symplectic flows (H.K.~Pak, "Transversal harmonic theory for transversally symplectic flows", J. Aust. Math. Soc. 84 (2008), 233--245) to the general transversely symplectic foliatio…
The paper examines metrics on foliated manifolds that have special geometric properties.
problem Characterizing metrics on foliated manifolds with specific harmonic properties.
method Examining the properties of bundle-like metrics on foliated manifolds.
result The interior product of basic harmonic forms is basic harmonic under certain conditions.
This paper investigates analytic properties of maps between hyperbolic surfaces, focusing on best Lipschitz maps and geodesic laminations.
problem Analyzing the properties of maps between hyperbolic surfaces, particularly best Lipschitz maps and their relationship to geodesic laminations.
method The authors produce best Lipschitz maps as limits of minimizers of p-Schatten integrals, addressing existence and regularity issues.
result The support of the measure dv, the derivative of a Lie algebra valued function v, lies on the canonical geodesic lamination constructed by Thurston.
In recent years a lot of attention has been paid to topological spaces which are a bit more general than smooth manifolds - orbifolds. Orbifolds are intuitively speaking manifolds with some singularities. The formal definition is also modelled on that of manifolds, an orbifold is a topological space which locally is ho…
We study basic Dolbeault cohomology and find new Weitzenböck formulas on a transversely Kähler foliation. We investigate conditions on mean curvature and Ricci curvature that impose restrictions on basic Dolbeault cohomology. For example, we prove that on a transversely Kähler foliation with positive transversal Ricci …
Study on twisted Dolbeault cohomology in Kähler foliations.
problem Exploring cohomology in transverse Kähler foliations.
method Analysis of twisted basic Dolbeault cohomology and transverse hard Lefschetz theorem.
result Proved Kodaira-Serre type duality for twisted basic Dolbeault cohomology.
The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article, similar results are obtained when the self-dual Weyl curvature is everywhere non-negat…
Overview of algebraic geometry for almost complex manifolds.
problem Developing algebraic geometry for almost complex manifolds without genericity.
method Reviewing results based on pseudoholomorphic maps and intersection theory.
result Introduction of birational morphism between almost complex manifolds.
In this article we study the linearized anisotropic Calderon problem. In a compact manifold with boundary, this problem amounts to showing that products of harmonic functions form a complete set. Assuming that the manifold is transversally anisotropic, we show that the boundary measurements determine an FBI type transf…
Study on biharmonic maps between conformally compact manifolds, proving non-existence under certain conditions.
problem Analyzing biharmonic maps on conformally compact manifolds.
method Investigating simple b-maps, focusing on non-existence results for biharmonic maps. result Non-existence of biharmonic maps under specific conditions, leading to implications for minimal surfaces.
Establishes jet transversality for regular maps from flexible manifolds.
problem Transversality for regular maps in algebraic geometry.
method Algebraic version of Forstnerič's theorem for holomorphic maps.
result Genericity theorems for regular maps of maximal ranks.
On a compact foliated Riemannian manifold with some transversal curvature conditions, there are no nontrivial basic harmonic forms (M. Min-Oo et al., J. Reine Angew. Math. 415 (1991). In this paper, we extend the above facts to a complete foliated Riemannian manifold.
The study examines harmonic symmetries on locally conformally Kähler manifolds, revealing properties of their kernels.
problem Characterizing harmonic symmetries on locally conformally Kähler manifolds.
method Analysis of harmonic differential forms and kernels of Laplacian-type operators.
result Properties of the kernel of certain Laplacian-type operators on locally conformally Kähler manifolds.
Maps to manifolds transverse to certain distributions satisfy an h-principle.
problem Maps to manifolds transverse to certain distributions.
method Proving the complete h-principle for transverse maps. result Maps to manifolds transverse to certain distributions satisfy the h-principle. We show how the tangent bundle decomposition generated by a system of ordinary differential equations may be generalized to the case of a system of second order PDEs `of connection type'. Whereas for ODEs the decomposition is intrinsic, for PDEs it is necessary to specify a closed 1-form on the manifold of independent …
We study transversality for Lipschitz-Fredholm maps in the context of bounded Fréchet manifolds. We show that the set of all Lipschitz-Fredholm maps of a fixed index between Fréchet spaces has the transverse stability property. We give a straightforward extension of the Smale transversality theorem by using the general…
New gauge preserves Einstein metrics' interactions, proving rigidity on negatively curved manifolds.
problem Stability and deformation theory of Einstein metrics.
method Introduces Chen-Nagano gauge condition, linking Lichnerowicz Laplacian to shifted scalar operator.
result Chen-Nagano gauge collapses to classical transverse-traceless gauge under spectral pinching assumptions.
Study on harmonicity of complex structure on product of trans-Sasakian manifolds.
problem Investigating harmonicity of complex structure on product of trans-Sasakian manifolds.
method Analysis of Levi-Civita connection and conditions for harmonicity on product manifold.
result Conditions for harmonicity of complex structure on product manifold of trans-Sasakian manifolds.
Study cohomology classes related to harmonic maps on submersions.
problem Understanding harmonic maps on submersions and their cohomology.
method Extending previous results on Riemannian submersions and p-harmonic morphisms to F-harmonic and f-harmonic maps.
result Extend results on Riemannian submersions to F-harmonic and f-harmonic maps.
We show a residues formula for maps generically transversal to regular holomorphic distributions.
The paper proves nonexistence of harmonic and bi-harmonic maps under specific conditions.
problem Existence of harmonic and bi-harmonic maps into certain Riemannian manifolds.
method Analysis of manifolds with conformal vector fields or Ricci solitons.
result Nonexistence of harmonic and bi-harmonic maps in specified conditions.
Dirac-harmonic maps are uncoupled under certain conditions.
problem Understanding the uncoupling of Dirac-harmonic maps.
method Critical points of a super-symmetric energy functional, with focus on harmonic maps.
result Dirac-harmonic maps are uncoupled under minimality assumption.
Extends p-harmonic map theory for new properties.
problem No specific problem stated; extends existing theory.
method Extended p-harmonic and biharmonic map definitions.
result New properties of generalized stable p-harmonic maps.
Analyzes harmonic and biharmonic maps from gradient Ricci solitons.
problem Characterizing maps from gradient Ricci solitons.
method Derives conditions for maps to be constant or harmonic.
result Biharmonic maps of finite energy from the two-dimensional cigar soliton are harmonic.
Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.
problem Understanding harmonic maps between singular spaces.
method Analyzing homogeneous harmonic maps between simplicial cones and their degrees.
result Degrees of homogeneous harmonic maps are related to eigenvalues of discrete graph Laplacians.
The paper examines Sasaki-Ricci solitons and their transverse rigidity properties.
problem Understanding the rigidity properties of Sasaki-Ricci solitons as singularity models.
method Established fundamental equations and criteria for transverse rigidity, proving key results about scalar curvature and Weyl tensor.
result Low-dimensional Sasaki-Ricci solitons with constant scalar curvature are Sasaki-Einstein, and those with harmonic Weyl tensor are finite quotients of the sphere.
Study heat flow for half-harmonic maps and harmonic maps with free boundary.
problem Integrability and regularity of half-harmonic maps and harmonic maps with free boundary.
method Introduced a heat flow associated to half-harmonic maps and constructed weak solutions via Ginzburg-Landau approximation.
result Proved partial regularity of weak solutions in space and time.
Study existence of harmonic and Dirac-harmonic maps from degenerating surfaces.
problem Existence of harmonic and Dirac-harmonic maps from degenerating surfaces.
method Using the Sacks and Uhlenbeck scheme, analyze a sequence of maps from degenerating surfaces to non-positive curved manifolds.
result Existence of limiting harmonic and Dirac-harmonic maps under certain conditions.
The paper constructs gluing maps for harmonic maps between Riemannian manifolds.
problem Constructing harmonic maps between Riemannian manifolds.
method Gluing construction of extended harmonic maps.
result Construction of gluing maps for harmonic maps under specific conditions.
The abstract discusses transversality for infinite dimensional manifolds.
problem Transversality for infinite dimensional Fréchet manifolds.
method Constructing degree of nonlinear Fredholm mappings using transversality results.
result Proof of rank theorem, invariance of domain theorem, and Bursuk-Ulam type theorem.
Both bi-harmonic map and f-harmonic map have nice physical motivation and applications. In this paper, by combination of these two harmonic maps, we introduce and study f-bi-harmonic maps as the critical points of the f-bi-energy functional 21∫Mf∣τ(φ)∣2dvg. This class of maps generalizes both …
∞-Harmonic maps are a generalization of ∞-harmonic functions. They can be viewed as the limiting cases of p-harmonic maps as p goes to infinity. In this paper, we give complete classifications of linear and quadratic ∞-harmonic maps from and into a sphere, quadratic ∞-harmonic maps between E…
We present a definable smooth version of the Thom transversality theorem. We show further that the set of non-transverse definable smooth maps is nowhere dense in the definable smooth topology. Finally, we prove a definable version of a theorem of Trotman which says that the Whitney (a)-regularity of a stratification…
Constructs harmonic maps between special geometric shapes.
problem Creating harmonic maps between specific types of geometric shapes.
method Equivariant harmonic maps constructed between cohomogeneity one manifolds.
result Developed a method to construct harmonic maps.
The paper proves a Liouville theorem for specific harmonic maps with free boundary.
problem Analyzing harmonic maps with free boundary conditions.
method Developed Liouville theorem for φ-F-symphonic, φ-F-harmonic, and φ-ΦS,p,ε harmonic maps. result Established Liouville theorem for the specified harmonic maps with free boundary.
Study cohomology classes related to n-harmonic morphisms and F-harmonic maps.
problem Understanding cohomology classes associated with n-harmonic morphisms and F-harmonic maps. method Utilizing the n-conservation law (2.6) to obtain sharp results. result Sharp results on cohomology classes related to n-harmonic morphisms and F-harmonic maps.