Study classifies 3D self-shrinkers with constant second form norm.
problem Classifying self-shrinkers with specific geometric properties.
method Analyzes 3D self-shrinkers in Euclidean space with constant second form norm.
result Classifies complete self-shrinkers with constant norm of the second fundamental form.
The study explores special surfaces in a normed space.
problem Constant Gaussian and mean curvature surfaces in normed spaces.
method Analyzes rotational surfaces with specific curvature properties.
result Generalizes catenoid, pseudo-sphere, and Delaunay surfaces.
Unique inhomogeneous ruled hypersurface found in complex hyperbolic space.
problem Classifying ruled real hypersurfaces with constant norm.
method Analyzing nonflat complex space forms, proving existence and uniqueness.
result Existence of a unique inhomogeneous example in complex hyperbolic space.
New method for efficient proximal mapping of 1-path-norm in shallow networks.
problem Efficiently handling the 1-path-norm of shallow neural networks.
method Closed-form proximal operator for efficient computation and upper bound on Lipschitz constant.
result Proximal mapping allows robust training against adversarial perturbations.
The paper proves inequalities linking geometric norms and Thurston norms in hyperbolic 3-manifolds.
problem Inequalities linking geometric norms and Thurston norms in hyperbolic 3-manifolds.
method Analyzes geometric L2-norms, Thurston norms, and Lipschitz maps to prove inequalities. result Proves an inequality between geometric L2-norm and Thurston norm, qualitatively sharp. A unique Kähler potential on the unit ball is identified with constant differential norm.
problem Finding a unique Kähler potential with constant differential norm on the unit ball.
method Analyzing the Kähler potential of the unit ball and its biholomorphic equivalence to the Siegel domain.
result The Kähler potential of the Siegel domain is unique up to automorphisms with constant differential norms.
Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and (-P). Then the polygon U, as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We define notions like Minkowskian curvature, evolutes and involutes for polygons of con…
Kernel interpolation is inconsistent for norms with smoothness above a constant.
problem Inconsistency of kernel interpolation in reproducing kernel Hilbert spaces.
method Lower bounds for generalization error in Sobolev norms.
result Kernel interpolation is always inconsistent for norms with smoothness above a constant.
The Hopf fibration is rigid among minimal maps between spheres.
problem Characterizing minimal submersions between spheres.
method Analyzing the properties of the Hopf fibration and minimal maps.
result The Hopf fibration is the only minimal submersion from S3 to S2 under certain conditions. Researchers classify special curved spheres in a complex space.
problem Classifying special holomorphic two-spheres in a complex Grassmannian.
method Completely classified noncongruent spheres with constant curvature and second fundamental form.
result Found all homogeneous spheres with constant curvature and second fundamental form.
Researchers classify 3D self-shrinkers in 4D space.
problem Classifying complete 3D self-shrinkers with specific properties in Euclidean space.
method Completely classified 3-dimensional complete self-shrinkers with constant norm of the second fundamental form and constant f3 in R4. result A complete classification of 3D self-shrinkers in Euclidean space R4. CNN layers with large norms are still robust to adversarial attacks.
problem Understanding the relationship between layer norms and adversarial robustness in CNNs.
method Theoretical analysis of ℓ1 and ℓ∞ norms, norm decay method, adversarial training frameworks. result Adversarially robust CNNs can have comparable or larger layer norms than non-adversarially robust ones.
We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature Kähler metric. This notion of uniform K-stability is analogous to coercivity of the Mabuchi functional. …
Sharp bounds on quasimode norms on compact space forms.
problem Characterize compact manifolds using quasimode decay rates.
method Analyzes upper and lower bounds of quasimode norms on compact space forms.
result Characterizes compact manifolds of constant curvature using quasimode decay rates.
Neural networks are dense among Lipschitz functions with fixed Lipschitz constant.
problem Characterizing neural network approximations to Lipschitz functions.
method Analyzing L-Lipschitz neural networks and their density in L-Lipschitz functions. result One layer neural networks are dense in the set of all L-Lipschitz functions. Improved bounds on curve filling areas in Banach spaces, leading to rigidity of Pu's inequality.
problem Improving bounds on curve filling areas in non-geodesic Banach spaces.
method Improved bounds on curve filling areas in Banach spaces.
result Rigidity of Pu's classical systolic inequality.
In this paper, we propose a method for estimating the Sobolev type embedding constant on a domain with minimally smooth boundary. We estimate the embedding constant by constructing an extension operator and computing its operator norm. We also present some examples of estimating the embedding constant for certain domai…
PSiLON Net uses L1 weight normalization and 1-path-norm regularization for efficient learning and sparsity.
problem Efficient learning and sparsity in neural networks with limited data.
method PSiLON Net employs L1 weight normalization and 1-path-norm regularization to simplify the 1-path-norm and achieve efficient learning and near-sparse parameters. result PSiLON Net achieves reliable optimization and strong performance in the small data regime.
On a closed balanced manifold, we show that if the Chern scalar curvature is small enough in a certain Sobolev norm then a slightly modified version of the Chern-Yamabe flow~\cite{Angella:2015aa} converges to a solution of the Chern-Yamabe problem. We also prove that if the Chern scalar curvature, on closed almost-Herm…
Improved eigenvalue bounds for minimal hypersurfaces in spheres.
problem Proving bounds on the first eigenvalue of minimal hypersurfaces in spheres.
method Using the Laplacian operator and properties of the second fundamental form, derived a new lower bound for the first eigenvalue.
result Improved lower bound for the first eigenvalue of minimal hypersurfaces in spheres.
Growth of spinors in 4D and 3D generalized Seiberg-Witten equations.
problem Proving growth of spinors in GSW equations on R4 and R3. method Unified framework of GSW equations, averaged L2-norm, curvature decay assumption, Yang-Mills-Higgs energy. result Growth of spinors in GSW equations on R4 and R3 faster than a power of the radius under suitable curvature decay. The paper studies scalar curvature of self-shrinkers and proves curvature bounds.
problem Proving curvature bounds for self-shrinkers.
method Analyzing scalar curvature of self-shrinkers in Euclidean space.
result Proves that the scalar curvature R of self-shrinkers is bounded by n−1. The paper analyzes the performance of empirical risk minimization for p-norm linear regression.
problem Empirical risk minimization on p-norm linear regression. method Analyzes performance under various conditions and moment assumptions.
result High probability excess risk bounds for empirical risk minimizer, matching asymptotic rates.
New method for differentially private optimization with general Lipschitz conditions.
problem Differentially private optimization under general Lipschitz conditions.
method Generalized Lipschitz condition for per-sample gradients, tuning clip norm based on minimum per-sample Lipschitz constant.
result Efficacy of the recommended clip norm tuning method verified on 8 datasets.
The paper bounds the L2-norm of Euler class for foliations on 3-manifolds.
problem Bounding the L2-norm of the Euler class for foliations on 3-manifolds. method Using constants bounding volume, radius of injectivity, sectional curvature, and mean curvature of leaves.
result Only finitely many cohomological classes can be realized by the Euler class of a transversely oriented foliation with bounded mean curvature.
We study the relationship between two norms on the first cohomology of a hyperbolic 3-manifold: the purely topological Thurston norm and the more geometric harmonic norm. Refining recent results of Bergeron, Şengün, and Venkatesh as well as older work of Kronheimer and Mrowka, we show that these norms are roughly propo…
In this paper we investigate complete critical metrics of the L2-norm of the scalar curvature. We prove that any complete critical metric with positive scalar curvature has constant scalar curvature and we characterize critical metrics with nonnegative scalar curvature in dimension three and four.
Study bounds on harmonic forms in hyperbolic 3-manifolds using Thurston norm and minimal surfaces.
problem Bounding the L2-norm of harmonic forms in hyperbolic 3-manifolds. method Using Thurston norm and interaction with minimal surfaces.
result Generalizes inequalities of Brock-Dunfield and studies sharpness in closed and cusped cases.
The paper examines biconservative hypersurfaces with constant curvature in space forms.
problem Characterizing biconservative hypersurfaces with specific curvature properties.
method Analyzing hypersurfaces with four distinct principal curvatures in space forms.
result Every biconservative hypersurface has constant mean and scalar curvature.
It is known that the L2-norms of a harmonic function over spheres satisfies some convexity inequality strongly linked to the Almgren's frequency function. We examine the L2-norms of harmonic functions over a wide class of evolving hypersurfaces. More precisely, we consider compact level sets of smooth regular…
Study complete 3D λ-translators in Minkowski space with constant properties.
problem Classify 3D space-like λ-translators with specific constant properties.
method Obtained classification theorem through analysis of constant norm and f4. result Classification theorem for 3D complete space-like λ-translators.
Let (Mn,g)(n≥3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and Rm˚ the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that Rm˚ goes to ze…
Let (M,g) be a noncompact complete n-manifold with harmonic curvature and positive Sobolev constant. Assume that L2 norms of Weyl curvature and traceless Ricci curvature are finite. We prove that (M,g) is Einstein if n≥5 and Ln/2 norms of Weyl curvature and traceless Ricci curvature are small enough…
Study on curvature conditions for non-conformally flat spheres using quasiconformal maps and Ricci flow.
problem Curvature conditions for non-conformally flat spheres.
method Construct quasiconformal maps and apply Ricci flow.
result Controlled bilipschitz constant between metrics.
Estimates the dual Thurston norm for foliations on negative curvature 3-manifolds.
problem Bounding the dual Thurston norm of foliations on 3-manifolds of negative curvature.
method Uses constants like injectivity radius, volume, curvature, and mean curvature of foliation leaves to estimate the dual Thurston norm.
result Provides an upper bound estimate on the dual Thurston norm of the Euler class of a foliation.
Equivalence of norms on manifolds with curvature bounds established.
problem Establishing equivalence of norms on manifolds with bounded sectional curvature.
method Using spectral projector and thickness condition for subsets.
result Constant in equivalence depends only on manifold dimension, curvature bounds, and frequency threshold.
In deep neural networks, the spectral norm of the Jacobian of a layer bounds the factor by which the norm of a signal changes during forward/backward propagation. Spectral norm regularizations have been shown to improve generalization, robustness and optimization of deep learning methods. Existing methods to compute th…
In this paper, we derive curvature estimates for strongly stable hypersurfaces with constant mean curvature immersed in Rn+1, which show that the locally controlled volume growth yields a globally controlled volume growth if ∂M=∅. Moreover, we deduce a Bernstein-type theorem for complete…
The paper classifies 2D complete Lagrangian self-expanders in complex 2-space.
problem Classifying complete Lagrangian self-expanders in complex 2-space.
method Obtained a classification theorem.
result A classification of 2D complete Lagrangian self-expanders with constant squared norm of the second fundamental form.
For complete Riemannian manifolds with vanishing Bach tensor and positive constant scalar curvature, we provide a rigidity theorem characterized by some pointwise inequalities. Furthermore, we prove some rigidity results under an inequality involving L2n-norm of the Weyl curvature, the traceless Ricci cur…
The study proves properties of self-shrinkers with bounded curvature.
problem Characterizing self-shrinkers with bounded curvature.
method Analyzing properties of self-shrinkers in Rn+1 with bounded second fundamental form. result Proves that if the squared norm of the second fundamental form is bounded, it must be constant.
In this paper, we give pinching Theorems for the first nonzero eigenvalue λ of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of M is 1 then, for any ε>0, there exists a constant C_ε depending on the dimension n of M and the L_∞-norm of the …
Given a normed plane P, we call P-cycloids the planar curves which are homothetic to their double P-evolutes. It turns out that the radius of curvature and the support function of a P-cycloid satisfy a differential equation of Sturm-Liouville type. By studying this equati…
We give improved algorithms for the ℓp-regression problem, minx∥x∥p such that Ax=b, for all p∈(1,2)∪(2,∞). Our algorithms obtain a high accuracy solution in O~p(m2p+∣p−2∣∣p−2∣)≤O~p(m31) iterations, where each iteration requires s…
In this paper we shall assume that the ambient manifold is a space form Nm+1(c) and we shall consider polyharmonic hypersurfaces of order r (briefly, r-harmonic), where r≥3 is an integer. For this class of hypersurfaces we shall prove that, if c≤0, then any r-harmonic hypersurface must be minima…
Proves inequality linking function deviation to gradient norm on compact manifolds.
problem Analyzing coupled elliptic systems on compact manifolds.
method Develops a new Poincaré-Sobolev inequality with a density-free reference average.
result Poincaré constant depends on the density's gradient norm.
We investigate the low-energy behavior of the gradient flow of the L2 norm of the Riemannian curvature on four-manifolds. Specifically, we show long time existence and exponential convergence to a metric of constant sectional curvature when the initial metric has positive Yamabe constant and small initial energy.
Geodesics in Kähler metrics connect metrics with constant scalar curvature.
problem Deriving geodesics for relatively Kähler metrics on fibrations.
method Deriving geodesic equation, proving uniqueness, convexity of log-norm functional.
result Fibrations with optimal symplectic connections are polystable.