Analyzes how minimal networks behave under boundary set deformations.
problem Behavior of minimal networks under boundary set deformations.
method Analytic deformations of boundary sets.
result Preservation of network types (minimal spanning trees, minimal fillings, stable shortest trees) under analytic deformations.
Researchers find multiple ways to deform manifolds with specific curvature properties.
problem Finding distinct conformal deformations of manifolds with boundary conditions.
method Using bifurcation results from Case, Moreira, and Wang, the researchers construct geometrically distinct solutions.
result There are multiple solutions to the conformal deformation problem in a finite set of dimensions.
Elliptic boundary value problem for G2 structures on manifolds.
problem Solving G2 holonomy equation on manifolds with boundary.
method Setting up a suitable linear elliptic boundary value problem.
result Existence of certain G2 cobordisms between deformations of Calabi-Yau 3-folds.
The paper studies curvature changes on manifolds with boundary.
problem Investigating conformal deformations of curvature on manifolds with boundary.
method Establishing sufficient conditions for positive scalar curvature and mean convex boundary, exploring further deformation scenarios.
result Conditions for conformal deformations to complete metrics with positive scalar curvature and mean convex boundary.
Study on deformations of special Lagrangians with boundary in Calabi-Yau manifolds.
problem Deformation problem for special Lagrangians with boundary constraints.
method Identifying tangent vectors with harmonic 1-forms vanishing on the boundary, proving unobstructed deformations.
result Moduli space of special Lagrangians with boundary is a smooth manifold.
Study on deforming calibrated submanifolds with boundary constraints.
problem Deforming calibrated submanifolds with boundary constraints in Riemannian manifolds.
method Extends McLean's deformation theory for closed compact submanifolds to include boundaries.
result Results extend McLean's theory to include boundaries, allowing for more flexible submanifold deformations.
The horocyclic deformation converges to the Gardiner-Masur boundary.
problem Understanding convergence of horocyclic deformations in Teichmüller space.
method Directed by a projective measured foliation, the deformation converges to the Gardiner-Masur boundary under certain conditions.
result The horocyclic deformation converges to the Gardiner-Masur boundary under specific conditions.
Develops a method to deform metrics on manifolds with non-compact boundaries.
problem Creating metrics with positive scalar curvature on manifolds with boundary.
method General deformation principle for Riemannian metrics on manifolds with non-compact boundaries.
result Non-existence of metrics with positive scalar curvature and mean convex boundary.
Simplicial sets deformation retract onto transverse simplices.
problem Deformation retraction of simplicial sets.
method Showed deformation retraction of singular simplicial set onto transverse simplices.
result Singular simplicial set deformation retracts onto transverse simplices.
Proof of theorem for 7D manifolds with boundary using Witten deformation.
problem Proving a theorem for 7-dimensional manifolds with boundary.
method Brute-force proof using Witten deformation.
result Proof of Kastler-Kalau-Walze type theorem for 7D manifolds with boundary.
The paper explores how to increase Steklov spectral gaps on manifolds with fixed boundary.
problem Finding ways to increase Steklov spectral gaps on manifolds with fixed boundary.
method Constructing compact manifolds with fixed boundary geometry and applying localized conformal deformations.
result It is possible to make the spectral gap arbitrarily large using localized conformal deformations.
Study on curvature functions for compact manifolds with boundary.
problem Understanding curvature functions on compact manifolds with boundary.
method Proves necessary and sufficient conditions for geodesic and Gaussian curvature, solves problems in the pointwise conformal case.
result New existence and nonexistence results for metrics with prescribed curvature, depending on Euler characteristic.
Developing deformation theory for Calabi-Yau 3-folds with boundary.
problem Dealing with Calabi-Yau threefolds on manifolds with boundary.
method Deformation theory and local Torelli Theorem for compact manifolds.
result An analogue of Hitchin's local Torelli Theorem for Calabi-Yau 3-folds with boundary, modulo a finite dimensional obstruction space.
New index formulae derived for operators on boundary groupoids.
problem Index theory on boundary groupoids of singular spaces.
method Deformation from pair groupoid and explicit construction of index map.
result Explicit index formulae for elliptic operators on boundary groupoids.
Study on deforming discrete conformal structures on surfaces with boundaries.
problem Deforming discrete conformal structures on surfaces with boundaries.
method Introduce combinatorial Ricci flow and combinatorial Calabi flow, establish longtime existence and global convergence of solutions.
result Effective algorithms for finding discrete hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.
Localized deformation of scalar curvature and mean curvature on manifolds.
problem Deforming scalar curvature and mean curvature on compact manifolds with boundary.
method Proving localized surjection of scalar curvature and mean curvature map, handling non-variational linearized problem.
result Localized deformations of scalar curvature and mean curvature on compact manifolds are possible.
The paper solves scalar curvature problems under conformal deformation for Riemannian manifolds.
problem Solving scalar curvature problems under conformal deformation for Riemannian manifolds.
method Pointwise conformal deformation, Yamabe equation with Dirichlet boundary conditions.
result Positive, smooth solutions to the Yamabe equation with Dirichlet boundary conditions.
Author reduces the Minkowski problem to the problem of construction the G-deformations preserving the product of principal curvatures for every point of surface in Riemannian space. G-deformation transfers every normal vector of surface in parallel along the path of the translation for each point of surface. The contin…
The paper studies how Kleinian groups can be deformed while preserving their peripheral structures.
problem Determining the extent of the island of discrete representations around the identity map in the space of representations.
method By cutting up the conformal boundary of a hyperbolic 3-manifold into a fundamental domain, the paper provides a computable region within which the fundamental domain is valid, ensuring peripheral structures remain similar under small deformations.
result The paper identifies a region in the space of representations of the fundamental group of a geometrically finite manifold, showing that groups in this region have peripheral structures that look coarsely similar.
Constructs obstructions and deformation principles for positive scalar curvature metrics with mean convex boundaries.
problem Obstructing the existence of positive scalar curvature metrics with mean convex boundaries.
method Atiyah-Patodi-Singer index formula, deformation principle, homotopy equivalences, higher homotopy groups.
result Construction of compact manifolds with nontrivial higher homotopy groups for positive scalar curvature metrics with mean convex boundaries.
On a manifold with boundary, we deform the metric conformally. This induces a deformation of the Schouten tensor. We fix the metric at the boundary and realize a prescribed value for the product of the eigenvalues of the Schouten tensor in the interior, provided that there exists a subsolution.
We consider natural conformal invariants arising from the Gauss-Bonnet formulas on manifolds with boundary, and study conformal deformation problems associated to them. The key technique we used is to derive boundary C^2 estimates directly from C^0 estimates for fully nonlinear equations. The main result has appeared i…
The problem of minimal distortion bending of smooth compact embedded connected Riemannian n-manifolds M and N without boundary is made precise by defining a deformation energy functional Φ on the set of diffeomorphisms $\diff(M,N)$. We derive the Euler-Lagrange equation for Φ and determine smooth minimizers o…
Smooth Riemannian manifolds can be embedded without boundary.
problem Embedding smooth Riemannian manifolds with boundary into complete manifolds without boundary.
method General gluing and conformal-deformation construction.
result Any smooth, metrically complete Riemannian manifold with smooth boundary can be realized as a closed domain into a smooth, geodesically complete Riemannian manifold without boundary.
Boundary value problems for operators of Dirac type arise naturally in connection with the conformal geometry of surfaces immersed in Euclidean 3--space. Recently such boundary value problems have been successfully applied to a variety of problems from computer graphics. Here we investigate under which conditions these…
Let M be a complete finite-volume hyperbolic 3-manifold with compact non-empty geodesic boundary and k toric cusps, and let T be a geometric partially truncated triangulation of M. We show that the variety of solutions of consistency equations for T is a smooth manifold or real dimension 2k near the point representing …
Proves a Schwarz-type lemma for noncompact manifolds with boundary.
problem Noncompact manifolds with boundary and their geometric applications.
method Suitable form of the weak maximum principle.
result Generalization of a classical result by Escobar.
Computes spectral Einstein functional for Witten deformation on even-dimensional spin manifolds.
problem Calculating the spectral Einstein functional for a specific deformation.
method Computes the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds.
result Computed the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds.
Study proves topological properties of isoperimetric sets in specific spaces.
problem Characterizing isoperimetric sets in PI spaces with deformation property.
method Proves topological regularity results using perimeter increment control.
result Isoperimetric sets are open, have boundary density estimates, and are bounded.
We study deformations of free boundary constant mean curvature (CMC) hypersurfaces whose Jacobi operator is degenerate due to symmetries of the ambient space. The value of the mean curvature and the ambient metric are allowed to vary simultaneously, provided that the infinitesimal ambient symmetries change smoothly. We…
Coassociative 4-folds are a particular class of 4-dimensional submanifolds which are defined in a 7-dimensional manifold M with a G_2 structure given by a `positive' differential 3-form, sometimes called G_2-form. Assuming that a G_2-form on M is closed, we study deformations of a compact coassociative submanifold N wi…
In this thesis we study the geometry of the fixed point set Σ of a smooth mapping Φ:M→M on a smooth compact Riemannian manifold M without boundary by computing the asymptotic expansion of the deformed heat trace $\Trace Φ\exp(tΔ)$ of the Laplace operator Δ on M. We assume that the fixed point set Σ is a…
Develops deformed algebras and groups from formal series over groupoids, with applications to cobordism.
problem Formal series over groupoids for algebras and groups.
method Deformation along a family of indexes, formal series.
result Regular Frölicher Lie groups and sometimes Fréchet Lie groups.
Given an ideal triangulation of a connected 3-manifold with non-empty boundary consisting of a disjoint union of tori, a point of the deformation variety is an assignment of complex numbers to the dihedral angles of the tetrahedra subject to Thurston's gluing equations. From this, one can recover a representation of th…
Locally connected deformation spaces for 3-manifolds.
problem Locating quasiconformally rigid points in hyperbolic 3-manifolds.
method Proving local connectedness at specific points in the deformation space.
result The deformation space is locally connected at quasiconformally rigid points.
Study the boundary of a submanifold of translation surfaces.
problem Understanding the boundary of a submanifold in a stratum of translation surfaces.
method Formula for tangent space to the boundary, finiteness results on cylinders, partial converse to Cylinder Deformation Theorem, generalizing part of Veech dichotomy.
result Formula for tangent space to the boundary of an affine invariant submanifold.
Symplectic coordinates found on projective structures on orbifolds.
problem Symplectic structure on deformation spaces of convex projective structures.
method Global Darboux coordinates system construction and symplectic space decomposition.
result Symplectic form on deformation space of convex projective structures.
Let M be an 8-manifold with a Spin(7)-structure. We first show that closed Cayley submanifolds of M form a smooth moduli space for a generic Spin(7)-structure. Then we study the deformations of a compact, connected Cayley submanifold X of M with non-empty boundary contained in a given submanifold W of M such that X and…
Let L be a special Lagrangian submanifold of a compact, Calabi-Yau manifold M with boundary lying on the symplectic, codimension 2 submanifold W. It is shown how deformations of L which keep the boundary of L confined to W can be described by an elliptic boundary value problem, and two results about minimal…
Study on ALH manifolds with boundary, showing surjectivity of scalar curvature map and mass rigidity.
problem Characterizing ALH manifolds with boundary and their mass.
method Scalar curvature deformation analysis and mass rigidity study.
result ALH manifolds that minimize mass integrals are characterized.
Deforms Seiberg-Witten solutions in 3D Lie group representations.
problem Non-compact moduli spaces of Seiberg-Witten solutions.
method Constructs Kuranishi models and discusses deformations.
result Fueter sections can be deformed to Seiberg-Witten solutions.
Paper extends noncompact Llarull's theorem to manifolds with boundary.
problem Finding upper bounds for scalar curvature infimum in noncompact manifolds.
method Using deformed Dirac operators to relax boundary conditions.
result Upper bound for scalar curvature infimum in terms of Laplacian spectrum.
The paper proves formulas and theorems for J-Witten deformation on specific manifolds.
problem Analyzing J-Witten deformation on specific types of manifolds.
method Obtained a Lichnerowicz type formula and proved Kastler-Kalau-Walze type theorems.
result Proved the Kastler-Kalau-Walze type theorems for J-Witten deformation on 4D and 6D almost product spin manifolds.
Study on metrics with positive scalar curvature and convex boundary.
problem Characterizing 3-manifolds with positive scalar curvature and convex boundary.
method Combination of earlier contributions, smoothing procedure, Ricci flow, and conformal deformation techniques.
result Path-connectedness of the moduli space of metrics.
We study the problem of conformal deformation of Riemannian structure to constant scalar curvature with zero mean curvature on the boundary. We prove compactness for the full set of solutions when the boundary is umbilic and the dimension n≤24. The Weyl Vanishing Theorem is also established under these hypothese…
Novel boundary conditions for Ricci flow to deform compact manifolds.
problem Deforming compact Riemannian manifolds with boundary using Ricci flow.
method Proposed boundary conditions that make first variations of functionals (Einstein-Hilbert action, lambda-functional) without boundary terms.
result Proof of short-term existence of solutions under proposed conditions.
The paper studies topological indices of geometric operators on manifolds with fibered boundaries.
problem Investigating indices of geometric operators on manifolds with fibered boundaries.
method Defining K-groups relative to pushforward for boundary fibration, using groupoid deformation techniques to prove properties of indices.
result Indices of twisted geometric operators can be understood as index pairings over K-groups.
New non-Kähler 3-folds constructed via log conifold transitions.
problem Constructing new non-Kähler 3-folds from Fano threefold pairs.
method Defining log conifold transitions and studying their deformation theory.
result Local smoothings of nodes can be lifted to global first-order deformations.