Stability of a new map derived from the equator map is analyzed.
problem Stability of a new map derived from the equator map.
method Detailed stability analysis of the generalized equator map as a critical point of the extrinsic k-energy and p-energy.
result Established generalizations of classical (in)stability results.
In this paper are studied the simplest patterns of axial curvature lines (along which the normal curvature vector is at a vertex of the ellipse of curvature) near a critical point of a surface mapped into R4. These critical points, where the rank of the mapping drops from 2 to 1, occur isolated in generic one parameter…
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.
The exponential map fails to be injective near critical points in sub-Riemannian geometry.
problem Injectivity failure of the exponential map at critical points in sub-Riemannian geometry.
method Analysis of the Hilbert invariant integral of the variational problem associated with the sub-Riemannian structure.
result Characterization of conjugate points in terms of metric structure.
Study on manifolds that map to lower dimensions with specific critical points.
problem Characterizing manifolds that map to Rn−1 with round fold maps. method Analyzing smooth n-dimensional closed manifolds with n≥4 and classifying round fold maps up to C∞ A--equivalence. result Determine which manifolds admit round fold maps into Rn−1 and classify these maps. The paper connects reflection groups to maps with specific dynamical properties.
problem Understanding the relationship between reflection groups and anti-rational maps.
method Established a correspondence using planar graphs.
result Complete answers to geometric mating problems for anti-rational maps.
Critical points of approximations of the Dirichlet energy à la Sacks-Uhlenbeck are known to converge to harmonic maps in a suitable sense. However, we show that not every harmonic map can be approximated by critical points of such perturbed energies. Indeed, we prove that constant maps and the rotations of S2 are th…
Study improves regularity estimates for harmonic maps into ellipsoids.
problem Independence of regularity estimates on harmonic maps with varying target dimensions.
method Analyzes harmonic maps into ellipsoids, uses Palais-Smale sequences, and critical metrics.
result Enhanced regularity estimates for Laplace harmonic eigenmaps.
Characterizes infinite harmonic maps using 1-currents.
problem Defines critical points of a non-differentiable functional.
method Uses subdifferential and geometric condition in terms of 1-currents.
result Geometric condition equivalent to criticality in terms of 1-currents.
Proves generalized Chen's conjecture for biharmonic maps on foliations.
problem Proves generalized Chen's conjecture for (F,F')-biharmonic maps.
method Analyzes (F,F')-biharmonic maps and their critical points.
result Proves generalized Chen's conjecture for (F,F')-biharmonic maps.
Noninjective monodromy found in polynomial critical point tracking.
problem Tracking critical points in polynomials leads to noninjective monodromy.
method Monic squarefree complex polynomials with prescribed critical point multiplicities.
result Monodromy map is noninjective for polynomials with exactly two critical points.
New conservation laws found for polyharmonic maps in critical dimension.
problem Existence of conservation laws for polyharmonic maps in critical dimension.
method Small perturbation of Uhlenbeck's gauge fixing matrix.
result Existence of conservation laws for elliptic systems of even order in critical dimension.
We study a functional, whose critical points couple Dirac-harmonic maps from surfaces with a two form. The critical points can be interpreted as coupling the prescribed mean curvature equation to spinor fields. On the other hand, this functional also arises as part of the supersymmetric sigma model in theoretical physi…
New findings on mapping class group actions on the circle, improving critical regularity.
problem Improving understanding of mapping class group actions on the circle.
method Analyzing actions of non-solvable groups and finite index subgroups of mapping class groups.
result Critical regularity of mapping class groups is at most one for surfaces of complexity at least three.
Examples of smooth maps with finitely many critical points in dimensions (4,3), (8,5) and (16,9)math.GT We consider manifolds M2n which admit smooth maps into a connected sum of S1×Sn with only finitely many critical points, for n∈{2,4,8}, and compute the minimal number of critical points.
Let f:Mm→Nn be a smooth map between two differential manifolds with N connected, f(M) closed and f(M)=N. In this short note, we show that either all the points of M are critical points of f or the dimension the collection of all critical points of f is not less than n−1. Some consequences of th…
Free maps exist on low-dimensional tori and closed surfaces.
problem Embedding closed surfaces in high-dimensional spaces.
method Factorization trick for constructing free immersions.
result Every closed surface embeds freely in \(\mathbb{R}^5\).
The paper studies harmonic maps between surfaces homotopic to a covering map, proving uniqueness and injectivity of Hopf differential.
problem Analyzing harmonic maps between surfaces in the homotopy class of a covering map.
method Proving the uniqueness of critical points and injectivity of Hopf differential for harmonic maps.
result The uniqueness of critical points of energy function and injectivity of Hopf differential are proven under specific conditions.
Proves transitivity of a specific class of quadratic polynomials.
problem Transitivity of pure Hurwitz classes of post-critically finite quadratic polynomials.
method Uses mapping classes of the sphere with finitely many marked points.
result Establishes transitivity for pure Hurwitz classes of post-critically finite quadratic polynomials.
The space of symplectic connections on a symplectic manifold is a symplectic affine space. M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms on this space is Hamiltonian and calculated the moment map. This is analogous to, but distinct from, the action of Hamiltonian diffeomorphism…
We compute the minimum number of critical points of a small codimension smooth map between two manifolds. We give as well some partial results for the case of higher codimension when the manifolds are spheres.
Motivated from the action functional for bosonic strings with extrinsic curvature term we introduce an action functional for maps between Riemannian manifolds that interpolates between the actions for harmonic and biharmonic maps. Critical points of this functional will be called interpolating sesqui-harmonic maps. In …
Novel Morse theory for mapping cone cohomology.
problem Cohomology of mapping cones varies with closed forms.
method Introduced a Morse complex for mapping cones.
result Cohomology of cone Morse complex is isomorphic to mapping cone cohomology.
Investigates energy minimizers and critical points of scale-invariant tangent-point energies for knots.
problem Finding and characterizing minimizers and critical points of scale-invariant tangent-point energies for closed curves.
method Develops convergence and regularity theories based on fractional Sobolev spaces and new energy functionals.
result Minimizing sequences converge to locally critical embeddings in all but finitely many points, and locally critical embeddings are regular.
For a finite-dimensional (but possibly noncompact) symplectic manifold with a compact group acting with a proper moment map, we show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, and that the set of critical points of the square of the moment map is a countable dis…
Study moment maps coupled with convex functions to find critical points.
problem Understanding critical points of moment maps coupled with convex functions.
method Develop a theory of moment maps coupled with an Ad_K-invariant convex function f on k*.
result Interpret Kähler-Ricci solitons as a special case of generalized extremal metrics.
Harmonic map flow's singularity properties proven with Lojasiewicz inequalities.
problem Finite-time singularities of harmonic map flow in critical dimensions.
method Proving a weighted Lojasiewicz inequality.
result Continuity of body map and no-neck property for bubble-tree decompositions.
Stability of biharmonic maps in critical dimension proven.
problem Stability of biharmonic maps between manifolds in critical dimension.
method Generalization of Morse stability theory to biharmonic maps, development of strong energy quantization method.
result Strong energy quantization in a wide class of problems in geometric analysis.
Study on metrics maximizing eigenvalues of Paneitz operator on 4-manifolds.
problem Investigating metrics maximizing eigenvalues of Paneitz operator.
method Critical points of eigenvalues of Paneitz operator on Riemannian metrics with fixed volume.
result Critical metrics associated with extrinsic conformal-harmonic maps into round spheres.
A Morse function f on a manifold with corners M allows the characterization of the Morse data for a critical point by the Morse index. In fact, a modified gradient flow allows a proof of the Morse theorems in a manner similar to that of classical Morse theory. It follows that M is homotopy equivalent to a CW-complex wi…
Maps of degree 1 and critical points on manifolds are studied.
problem Evaluate the minimal number of critical points on closed smooth manifolds.
method Examined maps of degree 1 and considered high-dimensional examples.
result In dimension 3 or less, CritM≥CritN for maps of degree 1. Study shows only rotations can be approximated by Ginzburg-Landau critical points.
problem Proving not all harmonic maps can be approximated by Ginzburg-Landau critical points.
method Rigidity theorem applied to Ginzburg-Landau energy critical points.
result Only rotations can be approximated by Ginzburg-Landau critical points.
3D manifolds can map to a plane with specific curve patterns.
problem Characterizing 3D manifolds that can map to the plane with certain curve patterns.
method Analyzing fold maps and their critical value sets.
result Closed orientable 3-manifolds admit round fold maps into the plane if and only if they are graph manifolds.
The paper tackles MAP inference over non-convex constraints in safety-critical settings.
problem Efficiently computing MAP predictions subject to non-convex constraints is challenging.
method The paper investigates conditions for exact and efficient MAP inference over continuous variables and devises scalable algorithms for both tractable and general cases.
result The proposed methods outperform constraint-agnostic baselines and scale to complex densities.
Given any n-tuple of complex numbers, one can canonically define a polynomial of degree n+1 that has the entries of this n-tuple as its critical points. In 2002, Beardon, Carne, and Ng studied a map θ:Cn→Cn which outputs the critical values of the canonical polynomial constructed from the…
When f : R power n to R power p, is a surjective real analytic map with isolated critical value, we prove that the (m)-regularity condition (in a sense we define) ensures that f ||f|| is a fibration on small spheres, f induces a fibration on the tubes and both fibrations are equivalent. In particular, we make the state…
The study examines critical points of a moment map for associative algebras.
problem Characterizing critical points of a moment map for associative algebras.
method Analyzes the moment map m for the action of GL(n) on Vn and studies critical points of the functional Fn. result Characterizes maxima, minima, and critical points of the functional Fn for associative algebras. In a recent paper~\cite{DDL10} we studied basic properties of partial immersions and partially free maps, a generalization of free maps introduced first by Gromov in~\cite{Gro70}. In this short note we show how to build partially free maps out of partial immersions and use this fact to prove that the partially free map…
In this paper, we consider critical maps of a horizontal energy functional for maps from a sub-Riemannian manifold to a Riemannian manifold. These critical maps are referred to as subelliptic harmonic maps. In terms of the subelliptic harmonic map heat flow, we investigate the existence problem for subelliptic harmonic…
We introduce Z-critical connections for holomorphic vector bundles and prove their existence under stability conditions.
problem Existence of Z-critical connections for holomorphic vector bundles. method Associated geometric PDEs to Bridgeland stability conditions and used infinite dimensional moment maps.
result In the large volume limit, a sufficiently smooth holomorphic vector bundle admits a Z-critical connection if and only if it is asymptotically Z-stable. Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.
problem Estimating the Morse index of anisotropic minimal surfaces.
method Local analysis of Gauss map, conformal geometric techniques applied to the Gauss map.
result Upper and lower estimates for the Morse index of anisotropic minimal surfaces.
Assume that there exists a smooth map between two closed manifolds Mm→Nk with only finitely many cone-like singular points, where 2≤k≤m≤2k−1. If (m,k)∈{(2,2),(4,3),(5,3),(8,5),(16,9)}, then Mm admits a locally trivial topological fibration over Nk and there exists a smooth map $…
Certain classes of 3-manifolds, following Thurston, give rise to a 'skinning map', a self-map of the Teichmüller space of the boundary. This paper examines the skinning map of a 3-manifold M, a genus-2 handlebody with two rank-1 cusps. We exploit an orientation-reversing isometry of M to conclude that the skinning map …
The normal map of curves is analyzed as a vector field on a cylinder.
problem Understanding the geometric properties of normal maps and their vector field interpretation.
method Interpreting critical points geometrically, studying Poincaré index, projecting to sphere, and analyzing winding and rotation indices.
result Counting theorems regarding winding and rotation indices of curves and their evolutes are proven.
The paper proves stability of critical points for conformally invariant Lagrangians.
problem Stability of critical points for conformally invariant Lagrangians under weak convergence.
method Upper-semi-continuity of Morse index plus nullity established for critical points.
result The sum of Morse indices and nullity is bounded from above by the sum of the Morse indices plus the nullity of the weak limit and bubbles.
In this paper, we prove that the Schrödinger map flows from Rd with d≥3 to compact Kähler manifolds with small initial data in critical Sobolev spaces are global. This is a companion work of our previous paper [23] where the energy critical case d=2 was solved. In the first part of this paper, for heat f…
In this paper we investigate the convergence properties of the upwards gradient flow of the norm-square of a moment map on the space of representations of a quiver. The first main result gives a necessary and sufficient algebraic criterion for a complex group orbit to intersect the unstable set of a given critical poin…
The Epstein deformation space parameterizes marked rational maps with prescribed combinatorial and dynamical structure. For the family of quadratic rational maps with a periodic critical cycle of order 4 and an extra critical point not lying in this cycle, S. Koch and I recently showed that the deformation space has in…