arXiv research

In this paper, we prove several formulas related to Hodge theory, and using them to prove the deformations of a compact $H$-twisted generalized Calabi-Yau manifold are unobstructed and $L^2$ convergence in a neighborhood in another power series . And if we assume that the deformation is smooth in a fixed neighborhood, …

We prove several formulas related to Hodge theory and the Kodaira-Spencer-Kuranishi deformation theory of Kähler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami differentials on Calabi-Yau manifolds and also a construction of global canonical family of ho…

Global fixed points in low-dimensional surface group space correspond to trivial representations.

Global homotopies upgrade classical map in differential geometry.

Local deformations of solutions to open PDEs can be extended globally if derivatives are constant along a subset.

The paper improves embedding results for $CR$ submanifolds, showing that compact deformations stay embeddable.

Singular fiber resolution does not describe the spontaneous breaking of gauge symmetry in F-theory, as the corresponding branch of the moduli space does not exist in the theory. Accordingly, even non-abelian gauge theories have not been fully understood in global F-theory compactifications. We present a systematic disc…

We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifolds, based on Kontsevich's local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection.

Study shows global invertibility in nonlinear elasticity with vanishing self-repulsion term.

New method tackles rugged optimization landscapes in contact-rich scenarios.

Study pinching sequences to understand degeneration of anti-de Sitter structures.

New non-Kähler 3-folds constructed via log conifold transitions.

A complex structure on a subset of S^6 cannot be extended to a global integrable structure.

The paper finds global Darboux coordinates for a new family of symplectic forms on the deformation space of $\mathbb{R}\mathbb{P}^2$-structures.

Global moduli theory for symplectic varieties proven.

Study on conformal deformations of complex Finsler metrics.

We show that any compact orientable hyperbolic 3-cone-manifold with cone angle at most πcan be continuously deformed to a complete hyperbolic manifold homeomorphic to the complement of the singularity. This together with the local rigidity by Hodgson and Kerckhoff implies the global rigidity for compact orientable hype…

Einstein manifolds are rigid under certain metric deformations.

Solves Calabi-Yau equation on symplectic manifolds using measurable Kahler metrics.

The paper proves a version of the SYZ conjecture for hyperkahler manifolds.

Characterizes neutral deformation modes of minimal surfaces.

Convex curves evolve into circles over time.

Symplectic coordinates found on projective structures on orbifolds.

Affine deformations of convex cones yield special spacetime structures.

The seven non euclidean geometries of the Thurston's geometrization program are proved to originate naturally from singularization morphisms and versal deformations on euclidean 3-manifolds generated in the frame of the Langlands global program. The Poincare conjecture for a 3-manifold appears as a particular case of t…

Global deformations of surfaces, immersed into the Euclidean 3-space, by using the modified Novikov--Veselov equation are investigated. relation to the theory of the Willmore functional is discussed

We consider some infinitesmal and global deformations of G_2 structures on 7-manifolds. We discover a canonical way to deform a G_2 structure by a vector field in which the associated metric gets "twisted" in some way by the vector cross product. We present a system of partial differential equations for an unknown vect…

We relate the global log canonical threshold of a variety with torus action to the global log canonical threshold of its quotient. We apply this to certain Fano varieties and use Tian's criterion to prove the existence of Kahler-Einstein metrics on them. In particular, we obtain simple examples of Fano threefolds being…

In this paper we describe the cohomogeneity one special Lagrangian 3-folds in the cotangent bundle of the 3-sphere, also known in the physics literature as a deformed conifold. Our main result gives a global foliation of the deformed conifold by T^2-invariant special Lagrangian 3-folds, where the generic leaf is topolo…

Paper proves stability of pseudo-Einstein contact form existence.

Representing 3D shape deformations by linear models in high-dimensional space has many applications in computer vision and medical imaging, such as shape-based interpolation or segmentation. Commonly, using Principal Components Analysis a low-dimensional (affine) subspace of the high-dimensional shape space is determin…

We study deformations of associative submanifolds $Y^3\subset M^7$ of a $G_2$ manifold $M^7$. We show that the deformation space can be perturbed to be smooth, and it can be made compact and zero dimensional by constraining it with an additional equation. This allows us to associate local invariants to associative subm…

Study of a series of Lorentzian structures on SL(2,R) with SO(1,1) symmetry.

We prove that the product of equators $S^{1} \times S^{1}$ in $S^{2} \times S^{2}$ is globally volume minimizing under Hamiltonian deformations.

Study on deforming discrete conformal structures on surfaces with boundaries.

Proves resurgent nature of a series solution to deformed Painlevé I equation.

Let $Σ$ be a connected, oriented surface with punctures and negative Euler characteristic. We introduce regular globally hyperbolic anti-de Sitter structures on $Σ\times \mathbb{R}$ and provide two parameterisations of their deformation space: as an enhanced product of two copies of the Fricke space of $Σ$ and as the b…

Continuous analysis techniques for deforming domains in manifolds.

The paper shows deformations between minimal surfaces in $S^{n+2}$ and $H^{n+2}$.

In this paper we deduce a local deformation lemma for uniform embeddings in a metric covering space over a compact manifold from the deformation lemma for embeddings of a compact subspace in a manifold. This implies the local contractibility of the group of uniform homeomorphisms of such a metric covering space under t…

We rigorously define the Liouville action functional for finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that the classical action - the critical point of the Liouville action functional, considered as a funct…

Unified framework connects deformation theory and derived categories for multiparameter persistence.

This paper studies global webs on the projective plane with vanishing curvature. The study is based on an interplay of local and global arguments. The main local ingredient is a criterium for the regularity of the curvature at the neighborhood of a generic point of the discriminant. The main global ingredient, the Lege…

Flow deforms locally convex curves into target curves.

In this paper, by using the Kuranishi coordinates on the Teichmüller space and the explicit deformation formula of holomorphic one-forms on Riemann surface, we give an explicit expression of the period map and derive new differential geometric proofs of the Torelli theorems, both local and global, for Riemann surfaces.

We give a global description of envelopes of geodesic tangents of regular curves in (not necessarily convex) Riemannian surfaces. We prove that such an envelope is the union of the curve itself, its inflectional geodesics and its tangential caustics (formed by the conjugate points to those of the initial curve along th…

Researchers compute curvatures of Stiefel manifolds with new metrics.

Solves generalized Kähler Calabi-Yau problem on compact manifolds.