arXiv research

Introduces new weighted floating functions and affine surface areas.

In this paper, we introduce a definition of $λ$-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that $λ$-hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete $λ$-hypersurfaces with …

In this paper we study the functional $\SW_{λ_1,λ_2}$, which is the the sum of the Willmore energy, $λ_1$-weighted surface area, and $λ_2$-weighted volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our main result is a complete classification of all …

Let $M$ be a weighted manifold with boundary $\partial M$, i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of the interior weighted area for deformations by hypersurfaces with boundary in $\p…

We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the unique compact, orientable, second order minima of the weighted area under variation…

Paper proves Harnack inequality for $f$-mean curvature flow.

Since $n$-dimensional $λ$-hypersurfaces in the Euclidean space $\mathbb {R}^{n+1}$ are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of complete $λ$-hypersurfaces. We give a gap theorem of complete $λ$-hypersurfaces with po…

Study proves no minimal surfaces can be contained in certain half-spaces or cones.

The paper proves weighted monotonicity theorems in various spaces and applies them to minimal surfaces.

New weighted surface area measures for convex bodies with applications.

The paper proves Calabi-Bernstein type results for minimal and maximal surfaces in 3D and 3D-L spacetime.

We prove two weighted geometric inequalities that hold for strictly mean convex and star-shaped hypersurfaces in Euclidean space. The first one involves the weighted area and the area of the hypersurface and also the volume of the region enclosed by the hypersurface. The second one involves the total weighted mean curv…

Paper finds relations between Willmore-type energies, weighted areas, and vertical potential energies for cylindrical critical points.

Asymptotic results for weighted floating bodies are established and used to obtain new proofs for the existence of floating areas on the sphere and in hyperbolic space and to establish the existence of floating areas in Hilbert geometries. Results on weighted best and random approximation and the new approach to floati…

The paper proves smoothness of almost-minimizers' boundaries near the free boundary.

The study connects minimal and maximal surfaces in 3D and 3-L space.

Enhances linear regression with Kalman filter for loss minimization.

We consider a conjecture made by Ge, Wang and Wu regarding weighted Alexandrov-Fenchel inequalities for horospherically convex hypersurfaces in hyperbolic space (a bound, for some physically motivated weight function, of the weighted integral of the $k^{\mathrm{th}}$ mean curvature in terms of the area of the hypersurf…

Adaptive loss function formulation is an active area of research and has gained a great deal of popularity in recent years, following the success of deep learning. However, existing frameworks of adaptive loss functions often suffer from slow convergence and poor choice of weights for the loss components. Traditionally…

Study on $L_p$ affine surface areas and their inequalities for convex bodies.

We prove that on a Riemannian manifold, a smooth differential form has a primitive with a given (functional) upper bound provided the necessary weighted isoperimetric inequalities implied by Stokes are satisfied. We apply this to prove a comparison predicted by Gromov between the cofilling function and the filling area…

New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.

We use a new approach that we call unification to prove that standard weighted double bubbles in $n$-dimensional Euclidean space minimize immiscible fluid surface energy, that is, surface area weighted by constants. The result is new for weighted area, and also gives the simplest known proof to date of the (unit weight…

Extends illumination bodies to non-Euclidean spaces and proves their volume derivative defines surface area.

Stable capillary surfaces in weighted balls are disks.

Hyperplanes, hyperspheres and hypercylinders in $\Bbb R^n$ with suitable densities are proved to be weighted minimizing by a calibration argument. Also calibration method is used to prove a weighted minimal hypersurface is weighted area-minimizing locally.

Study shows how certain metrics can be split into warped products.

We revisit the question of existence and regularity of minimizers to weighted least gradient problems on a fixed bounded domain, subject to a Dirichlet boundary condition, in the case where the boundary data is continuous and the weight function is C^2 and bounded away from zero. Under suitable geometric conditions on …

The paper extends rigidity results for $λ$-self-expanders to hyperplanes, spheres, and cylinders.

Study on Bayesian transformers finds issues with weight-space inference and prior specification.

Adapts AUM to identify ambiguous tasks in crowdsourced learning, improving generalization.

In this paper we describe a procedure for refining the given triangulation of a 3-manifold that scales the PL-metric according to a given weight function while creating no new normal surfaces. It is known that an incompressible surface $F$ in a triangulated 3-manifold $M$ is isotopic to a normal surface that is of mini…

New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.

New pseudometrics defined on knot spaces based on curve thickness and length.

A weighted area estimate for entire graphs with bounded weighted mean curvature in Gauss space is given by a simple proof. Bernstein type theorems for self shrinkers (\cite {wa}) as well as for graphic $λ$-hypersurfaces (\cite{ chwe2}) follow immediately as consequences.

In this paper we study the steepest descent $L^2$-gradient flow of the functional $\SW_{λ_1,λ_2}$, which is the the sum of the Willmore energy, $λ_1$-weighted surface area, and $λ_2$-weighted enclosed volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'.…

Proposes a new method to initialize neural networks by estimating global curvature of weights.

We study $λ$-hypersurfaces that are critical points of a Gaussian weighted area functional $\int_Σ e^{-\frac{|x|^2}{4}}dA$ for compact variations that preserve weighted volume. First, we prove various gap and rigidity theorems for complete $λ$-hypersurfaces in terms of the norm of the second fundamental form $|A|$. Sec…

We prove the quasimodularity of generating functions for counting pillowcase covers, with and without Siegel-Veech weight. Similar to prior work on torus covers, the proof is based on analyzing decompositions of half-translation surfaces into horizontal cylinders. It provides an alternative proof of the quasimodularity…

We formalize AURC and develop estimators for SC systems.

Algorithm selects variables and bandwidths for geographically weighted regression.

One reflection suffices for orthogonal weights, reducing GPU usage.

The paper studies stability and minimizing properties of higher codimensional surfaces in Euclidean space.

Paper discusses extending Gini score for tied rankings and case weights.

The paper shows how heat flow approximates area functional on specific geometric spaces.

Spheres minimize weighted curvature on spheres.

Proves strong Morse inequalities for area functional in low dimensions.

We study stable smooth solutions to the isoperimetric type problem for a Gaussian weight on Euclidean Space. That is, we study hypersurfaces $Σ^n \subset \mathbb R^{n+1}$ that are second order stable critical points of compact variations that minimize Gaussian weighted area and preserve Gaussian weighted volume. We sho…