Designing Bézier surfaces minimizing the L^2-norm of the Gaussian curvature

In freeform surface modelling, developable surfaces have much application value. But, in 3D space, there is not always a regular developable surface which interpolates the given boundary of an arbitrary piecewise smooth closed curve. In this paper, tensor product Bézier surfaces interpolating the closed curves are determined and the resulting surface is a minimum of the functional defined by the L2-integral norm of the Gaussian curvature. The Gaussian curvature of the surfaces is minimized by the method of solving nonlinear optimization problems. An improved approach trust-region form method is proposed. A simple application example is also given.

The Concavity of the Gaussian Curvature of the Convex Level Sets of p-Harmonic Functions with Respect to the Height

For the p-harmonic function with strictly convex level sets,we find an auxiliary function which comes from the combination of the norm of gradient of the p-harmonic function and the Gaussian curvature of the level sets of p-harmonic function.We prove that this curvature function is concave with respect to the height of the p-harmonic function.This auxiliary function is an affine function of the height when the p-harmonic function is the p-Green function on ball.

Gaussian Curvature Estimates for the Convex Level Sets of Solutions for Some Nonlinear Elliptic Partial Differential Equations

We give lower bound estimates for the Gaussian curvature of convex level sets of minimal surfaces and the solutions to semilinear elliptic equations in terms of the norm of boundary gradient and the Gaussian curvature of the boundary.

Liouville-Type Theorems for Conformal Gaussian Curvature Equations in R^2

In this paper, we derive an elementary identity for smooth solutions of the following equation:$$\Delta u\left( x \right) + K\left( x \right)e^{2u\left( x \right)} = 0\,{\rm in}\,R^2 $$and use it to get some global properties of the solutions.

An Isometric Embedding of Mbius Band with Positive Gaussian Curvature

The main purpose of this paper is to show that there is an isometric embedding from M?bius band to ?4 with positive Gaussian curvature.

A Note on Harnack Type Inequality for the Gaussian Curvature Flow

In this short note we present a new Harnack expression for the Gaussian curvature flow, which is modeled from the shrinking self similiar solutions. As applications we give alternate proofs of Chow’s Harnack inequality and entropy estimate.

紧黎曼曲面上锥度量的高斯博内公式

We prove a generalization of the classical Gauss-Bonnet formula for a conical metric on a compact Riemann surface provided that the Gaussian curvature is Lebesgue integrable with respect to the area form of the metric.We also construct explicitly some conical metrics whose curvature is not integrable.

From the second gradient operator and second class of integral theorems to Gaussian or spherical mapping invariants

By combining of the second gradient operator, the second class of integral theorems, the Gaussian-curvature-based integral theorems and the Gaussian （or spherical） mapping, a series of invariants or geometric conservation quantities under Gaussian （or spherical） mapping are revealed. From these mapping invariants important transformations between original curved surface and the spherical surface are derived. The potential applications of these invariants and transformations to geometry are discussed

On Fibre Bundle Surfaces and Their Curvatures

A surface model called the fibre bundle model is proposed. This model represents a surface locally as a direct product of two curves： a base curve and a fibre curve. We introduce the fibre bundle model and then obtain the Gaussian curvatures and the mean curvatures of a certain kind of fibre bundle surface models using 1-parameter groups of a linear Lie algebra as fibres. Some examples are given to verify our results.

A Vector Tensor Calculus Description of a Euclidean Space

We present a tensor description of Euclidean spaces that emphasizes the use of geometric vectors which leads to greater geometric insight and a higher degree of organization in analytical expressions. We demonstrate the effectiveness of the approach by proving a number of integral identities with vector integrands. The presented approach may be aptly described as absolute vector calculus or as vector tensor calculus.

CONSTRUCTION OF HOMOGENEOUS MINIMAL 2-SPHERES IN COMPLEX GRASSMANNIANS

In this paper, we construct a class of homogeneous minimal 2-spheres in complex Grassmann manifolds by applying the irreducible unitary representations of SU （2）. Furthermore, we compute induced metrics, Gaussian curvatures, Khler angles and the square lengths of the second fundamental forms of these homogeneous minimal 2-spheres in G（2, n ＋ 1） by making use of Veronese sequence.

An Existence Result of the Elliptic Equation Δu+K(x)e^(2u)=0

This paper considers the existence problem of an elliptic equation, which is equivalent to solving the so called prescribing conformal Gaussian curvature problem on the hyperbolic disc H^2. An existence result is proved. In particular, K（x） is allowed to be unbounded above.

On the Elliptic Equation △u+K(x)e^(2u)=0 with K(x) Positive Somewhere

This paper considers the existence problem of an elliptic equation, which is equivalent to the prescribing conformal Gaussian curvature problem on R^2. An existence result is proved. In particular, K（x） is allowed to be unbounded above.

Surfaces of Revolution in Minkowski 3-Space Satisfying F （G） = k（G ＋ C）

In this paper, we investigate the surfaces of revolution under the condition FIl（G） = k（G ＋ C）, where r11 is one of the Christoffel-like operators, G is the Gauss map of the surface, k is a non-constant function and C is a constant vector in Minkowski 3-space.

CONFORMAL DEFORMATION OF COMPLETE SURFACE WITH NEGATIVE CURVATURE

The author considers the problem of deforming the metric on a complete negatively curved surface conformal to another metric whose Gauss curvature is nonpositive.

Conformal minimal two-spheres in Q_n

In this paper we study,using moving frames,conformal minimal two-spheres S2 immersed into a complex hyperquadric Qn equipped with the induced Fubini-Study metric from a complex projective n+1-space CPn+1.Two associated functions τX and τY are introduced to classify the problem into several cases.It is proved that τX or τY must be identically zero if f:S2 → Qn is a conformal minimal immersion.Both the Gaussian curvature K and the Khler angle θ are constant if the conformal immersion is totally geodesic.It is also shown that the conformal minimal immersion is totally geodesic holomorphic or antiholomorphic if K = 4.Excluding the case K = 4,conformal minimal immersion f:S2 → Q2 with Gaussian curvature K2 must be totally geodesic with(K,θ) ∈ {(2,0),(2,π/2),(2,π)}.

Singularity radius gradient-based rapid singularity-escape steering law for SGCMGs

In order to visualize singularity of SGCMGs in gimbal angle space,a novel continuous bounded singularity parameter--Singularity Radius,whose sign can distinctly determine singularity type,is proposed.Then a rapid singularity-escape steering law is proposed basing on gradient of Singularity Radius and residual base vector to drive the SGCMG system to neighboring singular boundary,and quickly escape elliptic singularities.Finally,simulation results on Pyramid-type and skew-type configuration demonstrate the effectiveness and rapidness of the proposed steering law.

On Pseudo-Holomorphic Curves from Two-Spheres into a Complex Grassmannian G(2, 5)

Let s ： S2 → G（2, 5） be a linearly full totally unramified pseudo-holomorphic curve with constant Gaussian curvature K in a complex Grassmann manifold G（2, 5）. It is prove that K is either 1 4 1 or 4/5 if s is non-±holomorphic. Furthermore, K = 1/3 if and only if s is totally real. We also prove that the Gaussian curvature K is either 1 or -4/3 if s is a non-degenerate holomorphic curve under some conditions.

Completion of R^(2)with a Conformal Metric as a Closed Surface

In this paper,we obtain some asy mptotic behav ior results for solutions to the prescribed Gaussian curvature equation.Moreover,we prove that under a con-formal metric in R^(2),if the total Gaussian curvature is 4π,the conformal area of R^(2)is finite and the Gaussian curvature is bounded,then R^(2)is a compact C^(l,α)surface after completion at∞,for anya∈(0,1).If the Gaussian curvature has a Holder decay at in-finity,then the completed surface is C^(2).For radial solutions,the same regularity holds if the Gaussian curvature has a limit at infinity.