The paper proves geodesics and conic sections are length-minimizing under specific metrics.
problem Finding shortest paths in complex geometries.
method Calibrations and conformal metrics.
result Geodesics and conic sections are length-minimizing.
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48 results for “length-minimizing metrics”
Study on abnormal curves in sub-Riemannian manifolds, proving length-minimizing properties.
problem Characterizing abnormal geodesics in sub-Riemannian manifolds.
method Analyzing curves that annihilate Lie brackets and proving minimization properties.
result Strictly abnormal geodesics can cease to be locally length-minimizing.
Curvature bounds preserved in length-minimizing disks.
problem Preserving curvature bounds in length-minimizing disks.
method Analyzing length-minimizing disks and their curvature properties.
result Length-minimizing disks inherit curvature bounds from the target.
Geodesic currents on surfaces have comparable metrics in thick regions.
problem Comparing the geometry of geodesic currents and their minimizing metrics.
method Analyzing the space of geodesic currents on surfaces and comparing metrics on thick components.
result Geometries of geodesic currents and their minimizing metrics are comparable in thick regions.
Sharp proof of sub-Riemannian length-minimizing curves being at least
problem Smoothness of sub-Riemannian length-minimizing curves
method Study of a class of sub-Riemannian structures, proving regularity
result Theorem 1.1 in [6] is sharp
In this paper we consider the length minimizing properties of Hamiltonian paths generated by quasi-autonomous Hamiltonians on symplectically aspherical manifolds. Motivated by the work of L. Polterovich and M. Schwarz, we study the role of the fixed global extrema in the Floer complex of the generating Hamiltonian. Our…
The geodesic equation for the right invariant -metric (which is a weak Riemannian metric) on each Virasoro-Bott group is equivalent to the KdV-equation. We prove that the corresponding energy functional, when restricted to paths with fixed endpoints, has no local minima. In particular solutions of KdV don't define…
We prove the regularity for a class of abnormal length-minimizers in rank sub-Riemannian structures. As a consequence of our result, all length-minimizers for rank sub-Riemannian structures of step up to are of class .
We use the criteria of Lalonde and McDuff to determine a new class of examples of length minimizing paths in the group . For a compact symplectic manifold of dimension two or four, we show that a path in , generated by an autonomous Hamiltonian and starting at the identity, which induces no non-cons…
We prove that in a class of non-equiregular sub-Riemannian manifolds corners are not length minimizing. This extends the results [4]. As an application of our main result we complete and simplify the analysis in [6], showing that in a 4-dimensional sub-Riemannian structure suggested by Agrachev and Gauthier all length-…
We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parametrization of geodesic curves and the global existence of length minimizing geodesics are deduced using simple methods based on the calculus of varia…
The paper proves a nonholonomic version of Maupertuis-Jacobi principle and shows that nonholonomic trajectories minimize length.
problem Nonholonomic dynamics and their length minimization.
method Contact bundle formulation and geometric equivalence between problems.
result Regular solutions of nonholonomic mechanical problems are reparametrizations of geodesics with minimized Riemannian length.
Study shows not all smooth paths are optimal in certain geometric structures.
problem Existence of non-smooth sub-Riemannian minimizing geodesics.
method Constructed a but not length-minimizer example.
result Found a real-analytic sub-Riemannian structure with non-smooth minimizers.
Proves Goh conditions for singular curves with specific properties.
problem Finding optimal paths with specific geometric constraints.
method Proof of Goh conditions of order n and open mapping theorem.
result Establishes conditions for strictly singular curves of corank 1.
A projection maps geodesic currents to Teichmüller space.
problem Mapping geodesic currents to Teichmüller space.
method Equivariant, length-minimizing projection from filling currents to Teichmüller space.
result The projection is well-behaved and maps geodesic currents to Teichmüller space.
This paper is devoted to a third order study of the end-point map in sub-Riemannian geometry. We first prove third order open mapping results for maps from a Banach space into a finite dimensional manifold. In a second step, we compute the third order term in the Taylor expansion of the end-point map and we specialize …
We extend Thurston's metric to projective filling currents, embedding Teichmüller space into the larger space.
problem Embedding Teichmüller space into the space of projectivized filling currents.
method Extending the symmetrized Thurston metric to and studying its geometry.
result There is no quasi-isometric projection back from to .
A new Riemannian metric on curve spaces is complete and smooth.
problem Defining a complete metric on the space of embedded curves.
method Proposed a new Riemannian metric and proved its completeness.
result The proposed metric is complete in multiple senses.
Injectivity radius on Stiefel manifold is π.
problem Determining the injectivity radius of the compact Stiefel manifold.
method Utilized the property of geodesics being space curves of constant Frenet curvatures.
result The injectivity radius on the Stiefel manifold under the Euclidean metric is π.
Average signature measures geodesics in Lie groups.
problem Understanding geometric properties of Lie groups through geodesic paths.
method Introducing average signature and using it with trace operation to recover geometric properties.
result Average signature can recover geometric properties like dimension, diameter, volume, and scalar curvature.
The minimal area of Finsler disks with minimizing geodesics is at least 6/π r^2.
problem Finding the minimal area of Finsler disks with minimizing geodesics.
method Discretizing the Finsler metric using random geodesics and applying integral geometry formulas.
result The Holmes--Thompson area of Finsler disks with minimizing geodesics is at least 6/π r^2, with examples showing the inequality is sharp.
It is well-known that the class of piecewise smooth curves together with a smooth Riemannian metric induces a metric space structure on a manifold. However, little is known about the minimal regularity needed to analyze curves and particularly to study length-minimizing curves where neither classical techniques such as…
Study geodesics on infinite-dimensional manifolds using Finsler structures.
problem Geodesics on Fréchet manifolds of Riemannian metrics.
method Establish Riemann-Finsler structures, prove existence and minimality of geodesics, derive Euler-Lagrange equations.
result Geodesics on Fréchet manifolds of Riemannian metrics are length minimizing and satisfy Euler-Lagrange equations.
In this work, we establish new rigidity results for the Maslov class of Lagrangian submanifolds in large classes of closed and convex symplectic manifolds. Our main result establishes upper bounds for the minimal Maslov number of displaceable Lagrangian submanifolds which are product manifolds whose factors each admit …
This paper classifies geodesic orbit metrics on compact Lie group .
problem Classifying geodesic orbit metrics on compact Lie groups.
method Using representation theory of Lie subgroups, specifically weakly regular subgroups.
result Left-invariant geodesic orbit metrics on compact Lie group are classified.
In this paper, we use Floer theory to study the Hofer length functional for paths of Hamiltonian diffeomorphisms which are sufficiently short. In particular, the length minimizing properties of a short Hamiltonian path are related to the properties and number of its periodic orbits.
This paper, the second of a series, deals with the function space of all smooth Kähler metrics in any given closed complex manifold in a fixed cohomology class. The previous result of the second author \cite{chen991} showed that the space is a path length space and it is geodesically convex in the sense that any tw…
We give a short solution to one of the main open problems in subriemannian geometry. Namely, we prove that length minimizers do not have corner-type singularities. With this result we solve Problem II of Agrachev's list, and provide the first general result toward the 30-year-old open problem of regularity of subrieman…
Spirals are not shortest paths in certain sub-Riemannian geometries.
problem Nonminimality of spiral-like curves in sub-Riemannian manifolds.
method Construction of a competing curve to demonstrate non-minimality.
result Spiral-like curves are not length minimizing in sub-Riemannian manifolds.
Study integrability of geodesic flow on specific Lie groups.
problem Integrability of geodesic flow on metabelian nilpotent groups.
method Symplectic reduction procedure applied to sub-Riemannian geodesic flow on metabelian nilpotent groups.
result Showed integrability of normal Hamiltonian flow in Engel-type groups.
The paper analyzes discrete approximations to minimize curve length in Euclidean space.
problem Minimizing the length of curves between two sets in Euclidean space.
method Finite differences and numerical integration for discrete approximations.
result The squared length of the reconstructed curve converges to the squared minimal length with rate .
Study finds conjugate points in geodesics of Kolmogorov flows on torus.
problem Characterizing pairs of integers (m,n) for which geodesics have conjugate points.
method Analysis of geodesics in the group of volume-preserving diffeomorphisms of a torus using stream functions.
result Existence of conjugate points for all pairs of strictly positive integers (m,n).
We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-mi…
The group of diffeomorphisms of a compact manifold endowed with the L^2 metric acting on the space of probability densities gives a unifying framework for the incompressible Euler equation and the theory of optimal mass transport. Recently, several authors have extended optimal transport to the space of positive Radon …
In general relativity, spatial light rays of static spherically symmetric spacetimes are geodesics of surfaces in Riemannian optical geometry. In this paper, we apply results on the isoperimetric problem to show that length-minimizing curves subject to an area constraint are circles, and discuss implications for the ph…
We conjecture that a non-flat -real-dimensional compact Calabi-Yau manifold, such as a quintic hypersurface with D=6, or a K3 manifold with D=4, has locally length minimizing closed geodesics, and that the number of these with length less than L grows asymptotically as L^{D}. We also outline the physical arguments b…
On the number of geodesic segments connecting two points on manifolds of non-positive curvaturemath.DG
In this paper we show that on a complete Riemannian manifold of negative curvature and dimension every two points which realize a local maximum for the distance function are connected by at least geometrically distinct geodesic segments (i.e. length minimizing). Using a similar method, we obtain that in th…
The purpose of this article is to investigate the relationship between suborbifolds and orbifold embeddings. In particular, we give natural definitions of the notion of suborbifold and orbifold embedding and provide many examples. Surprisingly, we show that there are (topologically embedded) smooth suborbifolds which d…
In this paper we first show that the necessary condition introduced in our previous paper is also a sufficient condition for a path to be a geodesic in the group $\Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms. This applies with no restriction on . We then discuss conditions which guarantee that su…
Minimal networks on S^2math.DG
The paper extends minimal network theory to the sphere, proving local minimality.
problem Finding networks of minimal length on the sphere.
method Adapted spherical geometry, calibration method, and local metric perturbation estimates.
result Spherical minimal networks composed of great-circle arcs are locally length-minimizing within small geodesic balls.
Abstract framework for two meromorphic forms on punctured surfaces.
problem Developing a framework for two meromorphic forms on punctured Riemann surfaces.
method Abstract framework with Teichmüller regularity, degeneration detection, and pushability.
result Existence of a surface carrying two meromorphic differentials realizing any prescribed restricted pair.
Reduced sub-Riemannian time on a specific group structure.
problem Optimizing paths in a sub-Riemannian structure on a Carnot group.
method Proved conjectured cut times, compared with known results, and solved equations in elliptic functions.
result Reduced cut times for sub-Riemannian paths on the Cartan group.
Sub-Riemannian Geodesics on math.DG
Researchers find shortest paths on a special group structure.
problem Finding shortest paths on a specific group structure.
method Symmetry reduction to a simpler problem, then solving on a quotient space.
result Explicitly described length-minimizing geodesics.
CRC method provides tighter uncertainty intervals for CT images.
problem Expressing uncertainty in CT images in clinically meaningful terms.
method Semantically adaptive CRC procedure leveraging length minimization.
result Valid coverage of ground-truth images with tighter uncertainty intervals.
Minimal networks minimize length and mass in certain configurations.
problem Finding minimal networks that minimize length and mass.
method Global and local calibrations to prove minimization properties.
result Minimal networks minimize mass and interfaces in partitions.
This paper classifies and determines the length of the shortest filling pairs on a specific type of surface.
problem Classifying and determining the length of the shortest filling pairs on a specific type of surface.
method Classifying and determining the length of the shortest filling pairs on a specific type of surface.
result The paper classifies and determines the length of the shortest minimal filling pairs on a genus two surface.
Random curves on surfaces have predictable properties as they grow.
problem Characterizing topological properties of random curves on surfaces.
method Analyzing a simple random walk on the Cayley graph of surface groups.
result The properties of random curves are generic and predictable as they grow.
Consider the group $\Ham^c(M)$ of compactly supported Hamiltonian symplectomorphisms of the symplectic manifold $(M,\om)$ with the Hofer -norm. A path in $\Ham^c(M)$ will be called a geodesic if all sufficiently short pieces of it are local minima for the Hofer length functional $\Ll$. In this paper, we giv…