arXiv research

A simplified proof of the Alexander-Conway polynomial exists.

We give a geometric proof of existence of Whitney stratifications of definable sets in o-minimal structures.

We provide yet another proof of the existence of calibrated forecasters; it has two merits. First, it is valid for an arbitrary finite number of outcomes. Second, it is short and simple and it follows from a direct application of Blackwell's approachability theorem to carefully chosen vector-valued payoff function and …

We provide an alternative, simpler proof of the existence of thick triangulations for noncompact $\mathcal{C}^1$ manifolds. Moreover, this proof is simpler than the original one given in \cite{pe}, since it mainly uses tools of elementary differential topology. The role played by curvatures in this construction is also…

New methods prove existence of rotating shapes moving in space.

We give a simple proof of an extension of the existence results of Ricci flow of G.Giesen and P.M.Topping [GiT1],[GiT2], on incomplete surfaces with bounded above Gauss curvature without using the difficult Shi's existence theorem of Ricci flow on complete non-compact surfaces and the pseudolocality theorem of G.Perelm…

Paper proves regularity and existence of Riemannian splines.

After surveying existing proofs that every closed, orientable 3-manifold is parallelizable, we give three proofs using minimal background. In particular, our proofs use neither spin structures nor the theory of Stiefel-Whitney classes.

This is a sequel to the paper [Oh5] (or ArXiv:math.SG/0206092). The main purpose of the paper is to give the proof of an existence theorem, with energy bounds, of certain pseudo-holomorphic sections of the mapping cylinder that is needed for the proof of nondegeneracy of the homological invariant pseudo-norm which the …

In this short note, we give simple proof of the Ricci flow's local existence and uniqueness on closed Einstein manifolds. We suggest a new setting for studying the space of Riemannian metrics on a compact manifold.

We give a new proof of the isoperimetric inequality in the plane, based on Steiner's formula for the area of a convex neighborhood. This proof establishes the isoperimetric inequality directly, without requiring that we separately establish the existence of an optimal domain. In doing so, this proof bypasses the main d…

We present a short elementary proof of an existence theorem of certain CAT(-1)-surfaces in open hyperbolic 3-manifolds. The main construction lemma in Calegari and Gabai's proof of Marden's Tameness Conjecture can be replaced by an applicable version of our theorem.

Using Lipschitz distance on Outer space we give another proof of the train track theorem.

No fair and strategy-proof automated market maker exists for more than two assets.

Simplified proof classifies surfaces using normal curves.

Estimation is the computational task of recovering a hidden parameter $x$ associated with a distribution $D_x$, given a measurement $y$ sampled from the distribution. High dimensional estimation problems arise naturally in statistics, machine learning, and complexity theory. Many high dimensional estimation problems ca…

Simplified proof for Frank and Lieb's inequality on Heisenberg group.

Completes the proof of curvature tensor existence for Jacobi operators.

Simplified proofs and new distributions on anti-quasi-Sasakian manifolds.

Simple proof shows forecasts can be calibrated in a few periods.

We give a direct global proof for the existence of symplectic realizations of arbitrary Poisson manifolds.

In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, $C$, Andreev's Theorem provides five classes of linear inequalities, depending on $C$, for the dihedral angles, which are necessar…

Proof of existence and uniqueness of weighted Voronoi-Delaunay on polyhedral surfaces.

Proves existence and uniqueness of solutions for A_n tt*-Toda equations.

Simplified proofs for splitting homotopy idempotents.

Proof of Gromov's conjecture in 3D simplifies existing methods.

Kuranishi's proof of complex deformation theory revisited

Suppose $M$ is a compact n-dimensional manifold, $n\ge 2$, with a metric $g_{ij}(x,t)$ that evolves by the Ricci flow $\partial_tg_{ij}=-2R_{ij}$ in $M\times (0,T)$. We will give a simple proof of a recent result of Perelman on the non-existence of shrinking breather without using the logarithmic Sobolev inequality.

We give a variational proof of the existence and uniqueness of a convex cap with the given upper boundary. The proof uses the concavity of the total scalar curvature functional on the space of generalized convex caps. As a byproduct, we prove that generalized convex caps with the fixed boundary are globally rigid, that…

We give a new proof for the local existence of a smooth isometric embedding of a smooth $3$-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into $6$-dimensional Euclidean space. Our proof avoids the sophisticated arguments via microlocal analysis used in earlier proofs. In Part 1, we introduce …

New proof shows no Hölder embeddings into Heisenberg group.

We present a general existence proof for a wide class of non-linear elliptic equations which can be applied to problems with barrier conditions without specifying any assumptions guaranteeing the uniqueness or local uniqueness of particular solutions. As an application we prove the existence of closed hypersurfaces wit…

The paper proves the existence of a tubular neighborhood for Finsler submanifolds.

We use variational methods and a modified curvature flow to give an alternative proof of the existence of a self-shrinking torus under mean curvature flow. As a consequence of the proof, we establish an upper bound for the weighted energy of our shrinking doughnuts.

We provide a simpler proof and slight strengthening of Morrey's famous lemma on $\varepsilon$-conformal mappings. Our result more generally applies to Sobolev maps with values in a complete metric space and we obtain applications to the existence of area minimizing surfaces of higher genus in metric spaces. Unlike Morr…

Proof shows cones minimize certain geometric functionals.

An elementary proof found for the double bubble problem in a specific norm.

We provide a complete study of existence and uniqueness of solutions to the Lichnerowicz equation in general relativity with arbitrary mean curvature.

Uniform proof for Ricci flows on complete manifolds.

This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame which encompasses families of arrangements. The frame uses the notion of matroids. For the proof of the existence of the potentials, a power series ansatz is made. The proof that it works requires that ce…

We exploit four-dimensional tensor identities to give a very simple proof of the existence of a Lanczos potential for a Weyl tensor in four dimensions with any signature, and to show that the potential satisfies a simple linear second order differential equation, e.g., a wave equation in Lorentz signature. Furthermore,…

Lean Copilot uses LLMs to assist theorem proving in Lean, improving efficiency and automation.

We give a new proof of the existence of nontrivial quasimeromorphic mappings on a smooth Riemannian manifold, using solely the intrinsic geometry of the manifold.

We construct cup and cap products in intersection (co)homology with field coefficients. The existence of the cap product allows us to give a new proof of Poincare duality in intersection (co)homology which is similar in spirit to the usual proof for ordinary (co)homology of manifolds.

In 1969, Choquet-Bruhat and Geroch established the existence of a unique maximal globally hyperbolic Cauchy development of given initial data for the Einstein equations. Their proof, however, has the unsatisfactory feature that it relies crucially on the axiom of choice in the form of Zorn's lemma. In this paper we pre…

Proof shows homeomorphism problem is unsolvable.

A new simple proof for surface map degree inequality.

New proof of Lie algebroid action equivalence and integrability.