New proof of Willmore inequality using geometric divergence inequality.
problem Proving the Willmore inequality for bounded domains.
method Using a parametric geometric inequality derived from a divergence form geometric differential inequality.
result New proofs of quantitative Willmore-type and weighted Minkowski inequalities.
First geometric proof of the flyping theorem.
problem Proving Tait's flyping conjecture.
method Geometric proof using Greene's characterization, Menasco's crossing ball structures, and isotopy/re-plumbing moves.
result First entirely geometric proof of Menasco-Thistlethwaite's flyping theorem.
New geometric proof of convex function differentiability and approximation.
problem Second-order differentiability of convex functions and their approximations.
method Elementary geometric approach to prove classical and recent results.
result New proofs of Lusin approximation of convex functions and bodies by C1,1 functions. Introduces a new canonical connection for Riemannian manifolds and proves Frobenius theorem geometrically.
problem Geometric proof of the Frobenius theorem on Riemannian manifolds.
method Introduces a new canonical connection and applies it to prove the Frobenius theorem.
result Geometric proof of the Frobenius theorem.
We give an alternative proof of Madsen-Weiss' generalized Mumford conjecture. Our proof is based on ideas similar to Madsen-Weiss' original proof, but it is more geometrical and less homotopy theoretical in nature. At the heart of the argument is a geometric version of Harer stability, which we formulate as a theorem a…
Geometric proof shows topological invariance of handle homology.
problem Topological invariance of handle homology in manifolds.
method Entirely geometric proof using Cerf theory.
result Proof of ∂2=0 in chain complex defined by handle decomposition. We give a geometric proof of existence of Whitney stratifications of definable sets in o-minimal structures.
Geometrically constructs dilogarithm from Chern-Simons theory.
problem Dilogarithm function and its properties.
method Spin Chern-Simons invariant of C*-connections.
result Geometric proofs of dilogarithm identities and branching structure.
In this article, we give a geometric proof of the classification of complex vector cross product due to Lee-Leung.
Geometric proof of Regge symmetry in different geometries.
problem Regge symmetry in tetrahedra edge lengths.
method Simple geometric proof in Euclidean, spherical, and hyperbolic geometries.
result Verification of Regge symmetry across different geometries.
This paper has been withdrawn by the author due a crucial sign error in Theorem B. We present a geometric proof of Thom conjecture, which uses Khovanov homology. Our approach doesn't use any analytic methods and is quite different from proof given by Kronheimer and Mrowka in 1994.
Corollary 2.3 in our paper "A geometric proof of the Karpelevich-Mostow theorem", Bull. Lond. Math. Soc. 41 (2009), no. 4, 634-638, is false. Here we give a counterexample and show how to avoid the use of this corollary to give a simpler proof of Karpelevich-Mostow theorem. We also include a short discussion of the ori…
We give a proof that the geometric K-homology theory for finite CW-complexes defined by Baum and Douglas is isomorphic to Kasparov's K-homology. The proof is a simplification of more elaborate arguments which deal with the geometric formulation of equivariant K-homology theory.
New geometric proof for rational tangles links-quivers correspondence.
problem Recovering symmetric/antisymmetric colored HOMFLY-PT polynomials from a quiver.
method Geometric approach using winding numbers in punctured plane and its second configuration space.
result Explicit description of quivers for rational tangles.
Transformed geometry into algebra to prove Pick's theorem efficiently.
problem Translating geometric Pick's theorem into formal algebraic proof.
method Formalized geometric Pick's theorem into algebraic proof using Lean.
result Efficient formal proof of Pick's theorem.
Geometric proof of contractibility of unitary group in strong topology.
problem Contractibility of unitary group in strong operator topology.
method Direct geometric proof and construction of special subspaces and operators.
result Direct geometric proof of contractibility theorem.
New proofs for complex Hopf manifolds using geometric structures.
problem Proving properties of complex Hopf manifolds.
method Constructing integrable holomorphic G-structures and flat holomorphic Cartan geometries.
result Provided a new proof of flat holomorphic Cartan geometries on complex Hopf manifolds.
Proofs a theorem using basic geometric tools.
problem C2-rectifiability problem
method Elementary geometric measure theory and topology
result Gives a proof of Alberti's Luzin-type theorem
Proofs for Gauss map properties on surfaces with finite geometric type.
problem Properties of Gauss maps on surfaces with finite geometric type.
method Topological proof and generalization of the little Picard theorem.
result Gauss map can not omit three or more points for minimal and no flat surfaces.
In this note we give geometric formulations and proofs of three results of S. Morita. These results relate certain two dimensional cohomology classes of various moduli spaces of curves. We also give a geometric interpretation of a fourth result of Morita. One motivation of this work is to facilitate the application of …
Proof shows cones minimize certain geometric functionals.
problem Minimizing cones over spheres in geometric functionals.
method Proof by foliation analysis of cone leaves.
result Cone minimizes functionals for SkimesSl. We give a simple, direct proof of the backward uniqueness of solutions to a class of second-order geometric evolution equations including the Ricci and cross-curvature flows. The proof, based on a classical argument of Agmon-Nirenberg, uses the logarithmic convexity of a certain energy quantity in the place of Carleman…
Simplified proof of spin^c structures using twistor spaces.
problem Proving every compact oriented 4-manifold admits spin^c structures.
method Using twistor spaces to provide a simpler, more geometric proof.
result Simplified and clarified understanding of spin^c structures in 4-manifolds.
In this paper we give a geometric proof of the Karpelevich's theorem that asserts that a semisimple Lie subgroup of isometries, of a symmetric space of non compact type, has a totally geodesic orbit. In fact, this is equivalent to a well-known result of Mostow about existence of compatible Cartan decompositions.
Geometric approach to moment maps in complex geometry.
problem Constructing moment maps in complex geometry.
method Introducing universal families and equivariant differential forms.
result New geometric proofs and equations for moment maps.
Summary of tensor tomography proofs on manifolds with boundaries.
problem Proving injectivity of tensor tomography on compact Riemannian manifolds with boundaries.
method Summarized proofs from previous studies.
result Summary of proofs for s-injectivity.
Two new proofs classify complete totally geodesic subsets of complex hyperbolic plane.
problem Classify complete totally geodesic subsets of complex hyperbolic plane.
method Two new proofs: one algebraic and one geometric.
result Only complex geodesics and real planes are non-trivial complete totally geodesic subsets.
Proof shows volumes of certain geometric representations are always integers.
problem Integrality of volumes of specific geometric representations.
method Elementary, combinatorial-geometrical proof.
result Volumes of representations are integers when n≥2. A new simple proof for surface map degree inequality.
problem Degree of maps between closed surfaces.
method Elementary proof without additional techniques.
result A new proof of the inequality χ(M) ≤ d·χ(N).
A simple geometrical proof shows that any target function can be found in a random network's neighborhood.
problem Finding any target function in a random network's neighborhood.
method Geometrical proof using a simple model of a high-dimensional sphere projected onto a low-dimensional subspace.
result High-dimensional geometry ensures that a uniform distribution over a sphere reduces to a Gaussian distribution with negligible covariances, enabling the presence of any target function in a random network's neighborhood.
New proof confirms rolling objects can follow any path.
problem Existence of rolling objects following any path.
method Geometric proof for period-n trajectoids.
result Existence of period-n trajectoids for any smooth curve.
In this paper, we provide an essentially self-contained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of three-manifolds. In particular, we give a detailed exposition of a complete proof of the Poincaré con…
New proof and description of commutator subgroups for free and surface groups.
problem Understanding commutator subgroups of free and surface groups.
method Geometric proof and representation-theoretic description.
result New free generating sets and structure descriptions for commutator subgroups.
Constructs knots from 3-manifolds with specified geometric limits.
problem Build knots from 3-manifolds with specific geometric properties.
method Constructs explicit families of knots converging to specified 3-manifolds.
result Constructs knots from geometrically finite 3-manifolds with one end.
New proofs of geometric inequalities using Bochner formulas.
problem Geometric inequalities and mixed volumes in convex geometry.
method Reduction to Bochner formulas via spectral theorem.
result New, simpler proofs of Alexandrov-Fenchel and Alexandrov's inequalities.
New weighted geometric inequalities for hypersurfaces in R^n proved.
problem Proving new weighted geometric inequalities for hypersurfaces in R^n.
method Proof of a family of sharp weighted inequalities involving weighted k-th mean curvature integral and quermassintegrals.
result Generalization and new proof of Wei and Zhou's result without relying on earlier results.
Defines geometric quantization for non-compact Hamiltonian torus manifolds using index theory.
problem Geometric quantization for non-compact Hamiltonian torus manifolds.
method Deformation of Dirac operator along group orbits, localization to lattice points.
result Geometric quantization is independent of the choice of polarization.
New proof of Schwarzschild stability using geometric gauge.
problem Linear stability of Schwarzschild spacetime under gravitational perturbations.
method Employing a new geometric gauge and exploiting the structure of transport equations.
result Established both orbital and asymptotic stability for linearised quantities.
Proof of reverse isoperimetric inequality for black holes.
problem Reverse isoperimetric inequality for black holes in Einstein gravity.
method Geometric-analytic approach.
result Reversal of the usual isoperimetric inequality is explained by curved backgrounds governed by Einstein's equations.
We prove two theorems on the removal of singularities on the boundary of a pseudo-holomorphic curve. In one theorem, we need no apriori assumption on the area of the curve. The proof uses a doubling argument with the goal of converting curves with boundary to curves without boundary. Our method is new and geometric and…
Geometric proof shows complex endomorphisms have invariant axes.
problem Proving complex endomorphisms have invariant axes.
method Dual geometric proof using vector fields and bordism.
result Arbitrary complex endomorphisms have invariant axes.
We give a new proof of the fact that the condition of a Fano manifold admitting a Kähler-Einstein metric is Zariski-open (provided that the automorphism group is discrete). This proof does not use the characterisation involving stability. The arguments involve estimates of Futaki invariants obtained from a differential…
Optimal proof of finite small eigenvalues for specific geometric manifolds.
problem Proving finiteness of small eigenvalues for geometrically finite manifolds.
method Analyzing the spectrum of the Laplace operator on geometrically finite rank one locally symmetric manifolds.
result Optimal proof of finite small eigenvalues in a specific interval.
The Whitney-Graustein theorem states that regular closed curves in the 2-plane are classified, up to regular homotopy, by their rotation number. Here we give a simple proof based on contact geometry.
We use a simple geometric argument and small cancellation properties of link groups to prove that alternating links are non-trivial. This proof uses only classic results in topology and combinatorial group theory.
New proof finds three divergence-free vector fields for any 3D manifold.
problem Proving the existence of divergence-free vector fields on 3D manifolds.
method Using geometric properties of eigenspinors in three dimensions.
result Found three divergence-free vector fields that are orthogonal and have the same length at every point.
An elementary proof shows that quasi-isometric groups to integers are virtually integers.
problem Proving that quasi-isometric groups to integers are virtually integers.
method An elementary proof approach.
result Any finitely generated group quasi-isometric to the integers is virtually the integers.
The aim of this note is to give a geometric proof for classical local rigidity of lattices in semisimple Lie groups. We are reproving well known results in a more geometric (and hopefully clearer) way.