The paper discusses Nirenberg's work on geometric problems and his personality.
problem Geometric problems in mathematics.
method Method of moving planes and implicit fully nonlinear elliptic equations.
result Illustrates Nirenberg's contributions to geometric problems.
We compare the option pricing formulas of Louis Bachelier and Black-Merton-Scholes and observe -- theoretically as well as for Bachelier's original data -- that the prices coincide very well. We illustrate Louis Bachelier's efforts to obtain applicable formulas for option pricing in pre-computer time. Furthermore we ex…
Samuel J. Lomonaco Jr and Louis H. Kauffman conjectured that tame knot theory and knot mosaic theory are equivalent. We give a proof of the Lomonaco-Kauffman conjecture.
Topological obstructions to admissibility in σk-Loewner--Nirenberg problem
problem Admissibility condition for σk-Loewner--Nirenberg problem method Exhibit topological obstructions
result Illustrate with examples
Let (M,g) be a compact Riemannian manifold of dimension n \geq 2. In this work we prove the validity of the optimal L^p-Riemannian Gagliardo-Nirenberg inequality for 1 < p \leq 2. Our proof relies strongly on a new distance lemma which. In particular, we extend L^p-Euclidean Gagliardo-Nirenberg inequalities due to Del …
Proves existence and compactness of solutions to σ2-Nirenberg problem on sphere.
problem Existence and compactness of solutions to σ2-Nirenberg problem on S2. method Establishes Liouville type theorems, a priori estimates, and uses degree theory.
result Proves existence of at most one blow-up point for solutions to σ2-Nirenberg problem. New neural network solves Nirenberg problem for curvature on sphere.
problem Prescribing Gaussian curvature on S2 for metrics conformal to the round metric. method Mesh-free physics-informed neural network (PINN) that directly parametrises the conformal factor.
result Neural network achieves very low losses for realisable curvatures, distinguishing them from non-realisable ones.
In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with same exponent n(n>1), then it has exactly n-dimensional volume growth. As application, we obtain geometric and topological properties of Alexandrov space, Riemannian manifold …
Newlander-Nirenberg theorem extended to complex b-manifolds.
problem Characterizing complex b-manifolds.
method Involutive splitting of b-tangent bundle, formal local invariants, singular coordinate change.
result Complex b-manifolds have a single local model.
In this note, we study the curvature flow to Nirenberg problem on S2 with non-negative nonlinearity. This flow was introduced by Brendle and Struwe. Our result is that the Nirenberg problems has a solution provided the prescribed non-negative Gaussian curvature f has its positive part, which possesses non-degenera…
Verified numerics prove existence of a curvature solution with known symmetries.
problem Existence of a curvature solution for the Nirenberg problem.
method Verified numerics and computer assistance.
result Existence of a genuine solution with known symmetry groups.
This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine Lp−Sobolev inequalities. The logarithmic version of affine Lp−Sobolev inequalities is verified. Moreover, An alternative proof of the affine Mo…
The paper derives inequalities on Finsler manifolds, influenced by their curvatures.
problem Deriving inequalities on Finsler manifolds.
method Local and global geometric inequalities on Riemannian and Finsler manifolds.
result Generalized Caffarelli-Kohn-Nirenberg and Hardy type inequalities on Finsler manifolds.
We study a class of compact surfaces in R3 introduced by Alexandrov and generalized by Nirenberg and prove a compactness result under suitable assumptions on induced metrics and Gauss curvatures.
The paper analyzes high-dimensional sphere solutions to the Nirenberg problem with residual mass.
problem The Nirenberg problem on high-dimensional spheres with residual mass.
method Analysis of subcritical approximations and blowing up solutions.
result Comprehensive description of blowing up solutions, including blow-up points and rates.
We prove the concavity of p-Rényi entropy power for positive solutions to the doubly nonlinear diffusion equations on Rn or compact Riemannian manifolds with nonnegative Ricci curvature. As applications, we give new proofs of the sharp Lp-Sobolev inequality and Lp-Gagliardo-Nirenberg inequalities on…
Study on half spheres solves Nirenberg problem with complex blow-up analysis.
problem Prescribing scalar curvature on half spheres.
method Refined blow-up analysis of finite energy approximated solutions.
result Complex blow-up points and vortex problems reveal new connections.
Symmetry proven for positive solutions of a weighted p-Laplace operator inequality.
problem Proving symmetry of positive solutions to a specific type of inequality.
method Analyzing positive critical points of Caffarelli-Kohn-Nirenberg inequalities with a weighted p-Laplace operator.
result Complete classification and symmetry result for positive solutions in a range of parameters.
We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with the same exponent n≥3, then it has exactly the n-dimensional volume growth. As an application, if an n-dimensional Finsler manifold of non-negative n-Ricci curvature satisfies th…
We consider the fractional Nirenberg problem on the standard sphere Sn with n≥4. Using the theory of critical points at infinity, we establish an Euler-Hopf type formula and obtain some existence results for curvature satisfying assumptions of Bahri-Coron type.
This paper solves the Christoffel problem in hyperbolic space and its equivalent on spheres.
problem Prescribing curvatures for convex hypersurfaces in hyperbolic space.
method Proving a full rank theorem to establish the existence of solutions.
result Existence of solutions to the Christoffel problem and its equivalent Nirenberg-Kazdan-Warner problem on spheres.
We prove that n-dimensional (n⩾3) complete and non-compact metric measure spaces with non-negative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are close to the model metric measure n-space (i.e., the Euclidean metric n-space).
Generalizes complex manifolds to manifolds with corners and generalized corners.
problem Tackles the extension of complex structures to manifolds with corners and generalized corners.
method Uses complex structures on the b-tangent bundle and proves a formal Newlander-Nirenberg type theorem.
result Proves that along each corner stratum, the b-complex structure agrees with a standard model to infinite order.
ITF improves DSR but inflates curvature, while marginal likelihood reduces it, affecting QoIs.
problem Curvature mismatch between teacher forcing and marginal likelihood in chaotic dynamical systems.
method Comparing objective-induced curvatures of ITF and marginal likelihood in a probabilistic switching augmentation of AL-RNNs.
result Curvature inflation by ITF and reduction by marginal likelihood affect dynamical quantities of interest.
The paper extends Newlander-Nirenberg theorem to domains with C2 boundary.
problem Extending Newlander-Nirenberg theorem to domains with C2 boundary. method Analyzing formally integrable complex structures on domains with C2 boundary. result Existence of global holomorphic coordinate systems on the closure of a bounded strictly pseudoconvex domain.
Solves Loewner-Nirenberg problem on Riemannian manifolds for k ≤ n/2.
problem Solving the Loewner-Nirenberg problem on Riemannian manifolds for k ≤ n/2.
method Analyzes fully nonlinear Loewner-Nirenberg problem and uses conformal metrics.
result Solves the σk-Loewner-Nirenberg problem for all k ≤ n/2. In this paper, we prove that if a metric measure space satisfies the volume doubling condition and the Gagliardo-Nirenberg inequality with the same exponent n (n≥2), then it has exactly the n-dimensional volume growth. Besides, two interesting applications have also been given. The one is that we show that if…
New Cantor sets with high-dimensional projections discovered.
problem Understanding projections of Cantor sets in high dimensions.
method Construction and analysis of Cantor sets in Rn. result Cantor sets can be moved to have (n−2)-dimensional projections in (n−1)-planes. Proves existence of smooth metrics with specific curvature properties.
problem Existence of smooth metrics with prescribed negative Ricci curvature.
method Formulated and proved for general domains in Euclidean space.
result Existence of smooth complete conformal metrics with prescribed negative Ricci curvature.
In this work, we develop a study involving some nonlinear partial differential equations on spheres and hemispheres, with the zero Neumann boundary condition, which are so-called Brezis-Nirenberg type problems, and we give conditions on which such equations have only constant solutions. We also extend these results for…
Flow approach solves Ricci equation boundary problem.
problem Solving generalized Loewner-Nirenberg problem for σk-Ricci equation. method Flow approach to prove existence and uniqueness of solution.
result Solution converges to the boundary value as time goes to infinity.
The article resolves complex structures in transport twistor spaces, proving a Newlander-Nirenberg theorem.
problem Degenerate complex structures in transport twistor spaces.
method Holomorphic blow-down structure maps to resolve degeneracy and gain insight into complex geometry.
result Global and local β-maps for various metrics, proving a Newlander-Nirenberg theorem for degenerate complex structures.
Two flow methods solve a problem on Riemannian manifolds, proving convergence to the Loewner-Nirenberg solution.
problem Solving the Loewner-Nirenberg problem on compact Riemannian manifolds with boundary.
method Direct flow and Yamabe flow approaches.
result Convergence of the flows to the solution of the Loewner-Nirenberg problem under various conditions.
Study proves curvature prescription on spheres for k ≥ n/2.
problem Prescribing σk-curvature on standard spheres. method Existence and compactness theorems proved for k ≥ n/2.
result Extends Chang, Han, Yang's result for n = 4 and k = 2.
It is known that alternative links are pseudoalternating. In 1983 Louis Kauffman conjectured that both classes are identical. In this paper we prove that Kauffman Conjecture holds for those links whose first Betti number is at most 2. However, it is not true in general when this value increases, as we also prove by fin…
The Nirenberg problem yields multiple conformal metrics for a given scalar curvature function.
problem Finding multiple conformal metrics with a given scalar curvature on spheres.
method Morse theoretical methods and counting index formulae, leveraging subcritical approximation and blowing-up solutions.
result Arbitrarily many metrics can be found that are conformally equivalent to the standard sphere and have the desired scalar curvature.
After works by Michael and Simon [10], Hoffman and Spruck [9], and White [14], the celebrated Sobolev inequality could be extended to submanifolds in a huge class of Riemannian manifolds. The universal constant obtained depends only on the dimension of the submanifold. A sort of applications to the submanifold theory a…
The paper solves a Nirenberg problem on half spheres, finding multiple blow-ups.
problem Finding conformal metrics with prescribed scalar curvature and zero boundary mean curvature on half spheres.
method Constructing finite energy solutions to a subcritical approximation of the problem on half spheres of dimension \( n \geq 5 \).
result The solutions exhibit multiple blow-up of cluster-type at the same boundary point.
The paper proves existence of solutions to a Loewner-Nirenberg problem on Riemannian manifolds.
problem Existence of solutions to a specific nonlinear problem on Riemannian manifolds.
method Proves existence of viscosity solutions using approximating cones and limit of smooth solutions.
result Existence of a Lipschitz viscosity solution to the Loewner-Nirenberg problem.
This paper has been withdrawn by the author.
The biharmonic flow and Willmore flow are studied in higher dimensions using geometric evolution equations.
problem Prove global existence of the Willmore flow in higher dimensions.
method Apply Michael-Simon-Sobolev inequality and Gagliardo-Nirenberg inequalities to establish local energy estimates and maximal existence time.
result Global existence of the Willmore flow in higher dimensions is proven.
Paper proves a quantitative estimate for transforming almost complex structures into standard ones.
problem Transforming almost complex structures into standard ones on bounded domains.
method Proves existence of global diffeomorphisms under Hölder-Zygmund conditions.
result Existence of a global diffeomorphism in a specified Hölder-Zygmund class.
The study shows that close hypersurfaces have uniformly bounded inequalities.
problem Bounding inequalities for close hypersurfaces.
method Analyzing families of smooth hypersurfaces close to a fixed one.
result Uniformly bounded constants in Sobolev, Gagliardo-Nirenberg, and geometric Calderón-Zygmund inequalities.
The paper proves conditions under which certain geometric structures are rigid.
problem Local rigidity of manifolds with respect to Gagliardo-Nirenberg inequalities.
method Investigates local rigidity properties related to Gagliardo-Nirenberg constants and unweighted Yamabe-type constants.
result Conditions guaranteeing the flatness of manifolds under specific curvature conditions.
Solves geometric problems using fully nonlinear equations and Morse theory.
problem Geometric problems, specifically Loewner-Nirenberg and Yamabe problems.
method Investigates structure of fully nonlinear equations and applies Morse theory techniques.
result Constructs admissible metrics under weak conditions and demonstrates topological obstructions.
Maximal solution of a PDE shows boundary smoothness for certain domains.
problem Boundary behavior of solutions to a specific PDE.
method Reduction to a nonlinear Fuchsian elliptic PDE.
result Hyperbolic radius is smooth up to the boundary.
Solves nonlinear problems on metric structures through eigenvalue counting.
problem Nonlinear equations on metric structures
method Counting large eigenvalues of linearized operators
result Solves fully nonlinear Loewner-Nirenberg and Yamabe problems
Paper solves Christoffel-Minkowski problem in hyperbolic space.
problem Prescribing k-th horospherical p-surface area measure of h-convex domains in hyperbolic space. method Considered a fully nonlinear equation and used the full rank theorem with a viscosity approach.
result Existence of uniformly h-convex solution under appropriate assumptions.