New minimal surfaces found in a specific type of 3D space.
problem Finding minimal surfaces in a particular class of 3D spaces.
method Investigated minimal surfaces invariant under a specific group action in unimodular semidirect products.
result Described new examples of minimal surfaces in a specific 3D space.
Relations will be described between the quandle cocycle invariant and the minimal number of colors used for non-trivial Fox colorings of knots and links. In particular, a lower bound for the minimal number is given in terms of the quandle cocycle invariant.
New invariant prevents minimal submanifolds in curved spaces.
problem Preventing minimal submanifolds in curved spaces.
method Defined a conformal invariant and proved its application.
result Conformal invariant prevents minimal submanifolds for specific dimensions.
Empirical study of IRMv1, an invariant risk minimization framework.
problem Learning predictors invariant to spurious correlations across different training environments.
method Extending ColoredMNIST experiment to various settings.
result IRMv1 performs better as spurious correlation varies more widely.
Let (N,J) be a real 2n-dimensional nilpotent Lie group endowed with an invariant complex structure. A left-invariant Riemannian metric on N compatible with J is said to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics on (N,J) with the same scalar curvature. In…
Develops methods for computing conformal invariants of submanifolds.
problem Computing conformal invariants of submanifolds.
method Direct construction of extrinsic ambient space, global invariants of conformally compact minimal submanifolds, introduction of conformal submanifold scalars.
result Derives an explicit Gauss--Bonnet--Chern-type formula and proves a rigidity result.
It was proved by Graham and Witten in 1999 that conformal invariants of submanifolds can be obtained via volume renormalization of minimal surfaces in conformally compact Einstein manifolds. The conformal invariant of a submanifold Σ is contained in the volume expansion of the minimal surface which is asymptotic to $…
Counts minimal tori in Riemannian manifolds with 6 or more dimensions.
problem Counting minimal tori in Riemannian manifolds.
method Introduces a function to count minimal tori and shows invariance under metric perturbations.
result The count function is invariant under metric perturbations.
Infinite G-invariant minimal hypersurfaces found in Riemannian manifolds.
problem Finding minimal hypersurfaces in manifolds with group actions.
method New algorithm using multi-stage maximal cuttings.
result Each G-homology class admits infinitely many distinct realizations by embedded minimal G-hypersurfaces. IRM fails to capture natural invariances on simple problems.
problem IRM fails to capture natural invariances on simple problems.
method IRM formulation and practical linear variant.
result IRM can lead to worse generalization than unconstrained ERM.
Study delta invariant of minimal generic curves on rational surfaces.
problem Recover delta invariant of curve germs from surface singularity topology.
method Explicit formulae for minimal generic curves on rational surfaces, proving delta invariant values for quotient singularities.
result Explicit formulae and values for delta invariant of minimal generic curves on rational surfaces.
GALA framework learns invariant graph representations via environment augmentation with minimal assumptions.
problem Learning invariant graph representations from different environments without additional assumptions.
method Developed GALA framework with minimal assumptions of variation sufficiency and consistency. Uses an assistant model to differentiate graph environment changes.
result Extracting maximally invariant subgraphs to proxy predictions identifies underlying invariant subgraphs for successful out-of-distribution generalization.
Study invariant minimizers in convex functions under amenable groups.
problem Finding invariant minimizers in convex functions invariant under amenable groups.
method Analyze smallest closed invariant convex subsets and apply to invariant optimality problem.
result Clarifies relations between equivariant neural networks and statistical theorems.
In this short note, exploits of constructions of F-structures coupled with technology developed by Cheeger-Gromov and Paternain-Petean are seen to yield a procedure to compute minimal entropy, minimal volume, Yamabe invariant and to study collapsing with bounded sectional curvature on inequivalent smooth st…
A left invariant metric on a nilpotent Lie group is called minimal, if it minimizes the norm of the Ricci tensor among all left invariant metrics with the same scalar curvature. Such metrics are unique up to isometry and scaling and the groups admitting a minimal metric are precisely the nilradicals of (standard) Einst…
New minimal surfaces derived from helicoids.
problem Existence of minimal surfaces with specific symmetries.
method Balance equations and nodal limit analysis.
result Existence of new screw motion invariant minimal surfaces.
We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general type and minimal super-conformal surfaces. We prove a Bonnet-type theorem for st…
Minimal vector fields on oscillator groups studied, with specific conditions for minimality.
problem Characterizing minimal left-invariant unit vector fields on oscillator groups.
method Analyzing structure constants and harmonic maps into the unit tangent bundle.
result Minimal vector fields defined by specific conditions on oscillator groups.
We introduce Invariant Risk Minimization (IRM), a learning paradigm to estimate invariant correlations across multiple training distributions. To achieve this goal, IRM learns a data representation such that the optimal classifier, on top of that data representation, matches for all training distributions. Through theo…
Proposes IIB for domain generalization, overcoming failure modes of IRM.
problem Domain generalization with nonlinear classifiers and pseudo-invariant features.
method Invariant Information Bottleneck (IIB) using mutual information and variational formulation.
result Significantly outperforms IRM on synthetic datasets and real-world benchmarks.
Ellipsoids host infinitely many minimal tori, bifurcating from a 2-torus orbit.
problem Minimal tori in ellipsoids.
method Analyzing 3D ellipsoids invariant under a 2-torus action.
result Infinitely many distinct minimal tori bifurcate from a 2-torus orbit.
Let (N,J) be a simply connected 2n-dimensional nilpotent Lie group endowed with an invariant complex structure. We define a left invariant Riemannian metric on N compatible with J to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar…
We explicitly classify all S1-invariant free boundary minimal annuli and Möbius bands in Bn. This classification is obtained from an analysis of the spectrum of the Dirichlet-to-Neumann map for S1-invariant metrics on the annulus and Möbius band. First, we determine the supremum of the k-th normaliz…
Proposes an alternative invariance penalty to address domain generalization issues.
problem Addressing domain generalization problems by finding invariant representations.
method Revisits the Gramian matrix of the data representation to propose an alternative invariance penalty.
result The proposed approach guarantees recovery of an invariant representation under mild conditions.
Minimal orbits of semi-simple Lie groups are studied and related to invariant subspaces.
problem Characterizing minimal orbits of semi-simple Lie groups.
method Analyzing projective orbits induced by representations of semi-simple Lie groups and relating them to invariant subspaces of the underlying modules.
result Minimal orbits of semi-simple Lie groups are in bijection with minimal orbits of compact subgroups on invariant subspaces.
In this paper a study of G-minimality, i.e., minimality of four-manifolds equipped with an action of a finite group G, is initiated. We focus on cyclic actions on CP2#CP2, and our work shows that even in this simple setting, the comparison of G-minimality in the various categories, i.e., locally …
Invariant minimal surfaces in the real special linear group of degree 2 with canonical Riemannian and Lorentzian metrics are studied. Constant mean curvature surfaces with vertically harmonic Gauß map are classified.
In this note we define three invariants of contact structures in terms of open books supporting the contact structures. These invariants are the support genus (which is the minimal genus of a page of a supporting open book for the contact structure), the binding number (which is the minimal number of binding components…
The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4-manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fun…
We classify minimal hypersurfaces in Rn×Sm, n,m≥2, which are invariant by the canonical action of O(n)×O(m). We also construct compact and noncompact examples of invariant hypersurfaces of constant mean curvature. We show that the minimal hypersurfaces and the noncompact constant mean curvatu…
Minimal conformal foliations on Lie groups are shown to be fibres of harmonic morphisms.
problem Characterizing minimal conformal foliations on Lie groups.
method Analyzing left-invariant semi-Riemannian metrics and harmonic morphisms.
result Minimal conformal foliations of codimension two are fibres of complex-valued harmonic morphisms.
Paper proves finiteness and Morse index estimates for equivariant min-max hypersurfaces.
problem Existence and finiteness of G-invariant minimal hypersurfaces. method Equivariant min-max theory, compactness theorem, bumpy metrics theorem.
result Generalization of Morse index estimates to equivariant setting.
Squeezed knots are slices of minimal cobordisms; obstructions come from quantum knot invariants.
problem Characterizing and obstructing squeezed knots.
method Analysis of cobordisms, quantum knot invariants, and stable cohomology operations.
result Effective obstructions to squeezedness come from quantum knot invariants, notably Rasmussen invariant refinements.
Minimal discs count as knot invariants in hyperbolic 4-space.
problem Counting minimal discs with a given boundary in hyperbolic 4-space.
method Using J-holomorphic curves in twistor space to count minimal discs, considering singularities at infinity.
result The number of minimal discs with a given boundary is a knot invariant.
Generic smooth minimal hypersurfaces exist in 8D manifolds.
problem Existence of smooth minimal hypersurfaces in high-dimensional manifolds.
method Global perturbation argument and a novel geometric invariant.
result Generic metrics on 8D manifolds admit smooth minimal hypersurfaces.
Efficient cobordisms show minimal signature values on certain links.
problem Finding minimal signature values for specific link types.
method Constructing topological cobordisms between torus links and connected sums of trefoil knots.
result The signature invariant σω at ω=ζ6 takes minimal values on torus links. Study Froyshov invariants and closed geodesics in hyperbolic 3-manifolds.
problem Understanding Froyshov invariants in hyperbolic 3-manifolds.
method Analytic number theory and hyperbolic geometry.
result Effective upper bounds for Froyshov invariants and algorithm to compute them.
The paper studies special crystallizations of 4-manifolds to minimize certain PL-invariants.
problem Minimizing combinatorially defined PL-invariants in crystallizations of compact 4-manifolds.
method Analysis of semi-simple and weak semi-simple crystallizations to minimize regular genus, Gurau degree, gem-complexity, and trisection genus.
result An original theorem on the minimization of PL-invariants for compact 4-manifolds with weak semi-simple crystallizations.
We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field φ. For the normal case, we prove that a φ-invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a φ-invariant submanifold N everyw…
IRM fails to improve over standard methods in complex settings.
problem Learning invariant features for out-of-distribution generalization.
method Analysis of Invariant Risk Minimization (IRM) and related approaches under a general model.
result IRM can fail catastrophically in non-linear settings, even when test data are similar to training distribution.
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…
Minimal polynomial found for Riemannian C_0-spaces.
problem Understanding the structure of Riemannian C_0-spaces.
method Constructing polynomial functions on tangent spaces and gluing them globally.
result The degree of the polynomial provides an upper bound for the Singer invariant.
The study proves a generic multiplicity one theorem for G-invariant minimal hypersurfaces.
problem Proving a generic multiplicity one theorem for G-invariant minimal hypersurfaces. method Equivariant min-max theory and analysis of G-homology classes. result Shows a generic multiplicity one theorem for G-invariant minimal hypersurfaces. Study on Lie groups' conformal foliations and harmonic morphisms.
problem Characterizing minimal conformal foliations on Lie groups.
method Analyzing left-invariant metrics and foliations on Lie groups.
result Minimal conformal foliations on Lie groups are fibres of harmonic morphisms.
Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.
problem Finding sharp lower bounds for modular invariants and Dehn twist coefficients.
method Analyzing the relation between fractional Dehn twists and modular invariants, classifying pseudo-periodic maps, and proving rigidity properties.
result Sharp lower bounds for modular invariants and Dehn twist coefficients in genus 2 and 3.
The paper proves the existence of G-invariant minimal hypersurfaces on certain Riemannian manifolds.
problem Existence of G-invariant minimal hypersurfaces on specific Riemannian manifolds. method Adapted Almgren-Pitts min-max theory to a G-equivariant version. result Existence of nontrivial closed smooth embedded G-invariant minimal hypersurfaces. Some elementary considerations are presented concerning Catenoids and their stability, separable minimal hypersurfaces, minimal surfaces obtainable by rotating shapes, determinantal varieties, minimal tori in S3, the minimality in Rnk of the ordered set of k orthogonal equal-length n-vectors, and U(1)-invariant minimal…
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.