Prescribing curvatures on surfaces with conical singularities and corners

Conformal metrics with prescribed gaussian and geodesic curvatures

This paper is concerned with the compactness of metrics of the disk with prescribed Gaussian and geodesic curvatures. We consider a blowing-up sequence of metrics and give a precise description of its asymptotic behavior. In particular, the metrics blow-up at a unique point on the boundary and we are able to give necessary conditions on its location. It turns out that such conditions depend locally on the Gaussian curvatures but they depend on the geodesic curvatures in a nonlocal way. This is a novelty with respect to the classical Nirenberg problem where the blow-up conditions are local, and this new aspect is driven by the boundary condition.

We study the Berger-Nirenberg problem on surfaces with conical singularities, i.e. we discuss conditions under which a function on a Riemann surface is the Gaussian curvature of some conformal metric with a prescribed set of singularities of conical types.

We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with the Neumann boundary condition. In the previous paper, we show that the Chern-Simons Higgs equation with parameter λ > 0 has at least two solutions (uλ1, uλ2) for λ sufficiently large, which satisfy that uλ1 → −u0 almost everywhere as λ → ∞, and that uλ2 → −∞ almost everywhere as λ → ∞, where u0 is a (negative) Green function on M. In this paper, we study the asymptotic behavior of the solutions as λ → ∞, and prove that uλ2 − \(\overline {u_\lambda ^2 } \) converges to a solution of the Kazdan-Warner equation if the geodesic curvature of the boundary ∂M is negative, or the geodesic curvature is nonpositive and the Gauss curvature is negative where the geodesic curvature is zero.

Weighted barycentric sets and singular Liouville equations on compact surfaces

Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities

From the Aharonov-Bohm effect to general relativity, geometry plays a central role in modern physics. In quantum mechanics, many physical processes depend on the Berry curvature. However, recent advances in quantum information theory have highlighted the role of its symmetric counterpart, the quantum metric tensor. In this paper, we perform a detailed analysis of the ground state Riemannian geometry induced by the metric tensor, using the quantum $XY$ chain in a transverse field as our primary example. We focus on a particular geometric invariant, the Gaussian curvature, and show how both integrals of the curvature within a given phase and singularities of the curvature near phase transitions are protected by critical scaling theory. For cases where the curvature is integrable, we show that the integrated curvature provides a new geometric invariant, which like the Chern number characterizes individual phases of matter. For cases where the curvature is singular, we classify three types (integrable, conical, and curvature singularities) and detail situations where each type of singularity should arise. Finally, to connect this abstract geometry to experiment, we discuss three different methods for measuring the metric tensor: via integrating a properly weighted noise spectral function or by using leading-order responses of the work distribution to ramps and quenches in quantum many-body systems.

A theorem of R.Hamilton [6] and B.Chow states that for every initial metric the initial value problem (1) possesses a unique solution which is defined for all t ≥ 0. Moreover, for t → ∞ the solution converges exponentially to a metric of constant Gauss curvature. Different proofs for this result were given by J.Bartz, M.Struwe and R.Ye [2] and by M.Struwe [13]. On a surface with boundary, one can demand that the Gauss curvature in the interior is constant and the geodesic curvature at the boundary vanishes. Alternatively, one can require that the geodesic curvature at the boundary is constant and the Gauss curvature in the interior vanishes. The evolution equation corresponding to the first case is

From the position of pattern making, we searched for the relation between plane development and 3D torso surface curvature by examining angle at concentrated vertices. This method will be referred to as the “Angle at vertex method”. The body forms of 110 young women were recorded with a 3D digitizer, and simulated by a triangulated polyhedron. Each concentrated Angle θ at the vertex was calculated for either internal Concentrated Gaussian Curvature (2π-θ), or boundary line Concentrated Geodesic Curvature (7π-θ). The total of Concentrated Sum of Gaussian Curvature and Geodesic Curvature for each model was -4π. A pattern of angular conservation was established, in accordance with Gauss-Bonnet theorem. On a macro level, we evaluated the distribution of curvature and general features of curvature for each model. On a micro level, we examined the regional characteristics in terms of sum positive values and sum negative values independently for both curvatures types, using the simulated Standard Model. The areas of the bust and shoulder will be difficult to construct from plan due to the positive and negative Gaussian Curvatures. Geodesic Curvature is concave around the neck-base line and armscye line, and the waist line is almost straight when translated to plan. The curvatures of every point on the Standard Model were color displayed, making it a highly informative tool for understanding the torso surface form.

We study the rst nonzero p-Steklov eigenvalue on a two-dimensional compact Riemannian manifold with a smooth boundary along the geodesic curvature ow.We prove that the rst nonzero p-Steklov eigenvalue is nondecreasing if the initial metric has positive geodesic curvature on boundary M and Gaussian curvature is identically equal to zero in M along the un-normalized geodesic curvature ow.An eigenvalue estimation is also obtained along the normalized geodesic curvature ow.

We investigate the logarithmic convexity of the length of the level curves for harmonic functions on surfaces and related isoperimetric type inequalities. The results deal with smooth surfaces, as well as with singular Alexandrov surfaces (also called surfaces with bounded integral curvature), a class which includes for instance surfaces with conical singularities and surfaces of CAT(0) type. Moreover, we study the geodesic curvature of the level curves and of the steepest descent for harmonic functions on surfaces with non-necessarily constant Gaussian curvature K. Such geodesic curvature functions turn out to satisfy certain Laplace-type equations and inequalities, from which we infer various maximum and minimum principles. The results are complemented by a number of growth estimates for the derivatives L' and L'' of the length of the level curve function L, as well as by examples illustrating the presentation. Our work generalizes some results due to Alessandrini, Longinetti, Talenti, Ma–Zhang and Wang–Wang.

The interplay between partial differential equations (PDEs) and the theory of mappings has a long and distinguished history, and that connection underpins this book. Gauss’s practical geodesic survey work stimulated him to develop the theory of conformal transformations, for mapping figures from one surface to another. For conformal transformation from plane to plane he used a pair of equations apparently derived by d’ Alembert, who first related the derivatives of the real and imaginary part of a complex function in 1746 in his work on hydrodynamics (31], p. 497. These equations have become known as the Cauchy–Riemann equations. Gauss developed the differential geometry of surfaces around 1827, emphasizing the intrinsic geometry, with Gaussian curvature defined by measurements within the surface. If a surface is deformed conformally (preserving angles), then the Gaussian curvature is unchanged, and hence the intrinsic geometry of the surface is unaffected by such deformations. Gauss also considered geodesic curves within surfaces. In 1829 Lobachevsky constructed a surface (the horosphere) within his non-Euclidean space, such that the intrinsic geometry within that surface is Euclidean, with geodesic curves being called Euclidean lines. For the converse process, he could only suggest tentatively that, within Euclidean space, the intrinsic geometry of a sphere of imaginary radius was Lobachevskian. But imaginary numbers were then regarded with justifiable suspicion, and he did not propose that as an acceptable model of his geometry within Euclidean space. In his most famous work, Beltrami [32] showed that Lobachevsky’s geometry is the intrinsic geometry of a surface of constant negative curvature, with geodesic curves being called lines in Lobachevsky’s geometry. Beltrami illustrated various surfaces with constant negative curvature, the simplest of which is the pseudosphere generated by revolving a tractrix around its axis. Beltrami’s paper convinced most mathematicians that the geometries of Euclid and of Lobachevsky are logically equivalent. In that work Beltrami used a differential equation corresponding to Gauss’s equation. This has come to be known as Beltrami’s equation, and later in this book we shall present the most recent developments in this area, solving Beltrami’s equation at the critical point. where uniform ellipticity bounds are lost. This will necessitate the development of some considerable technical machinery to enable us to move away from the classical setting of uniformly elliptic PDEs to the case of degenerate elliptic equations. Beltrami’s equation and its solutions, the quasiconformal mappings, have found a home in virtually all aspects of modern complex analysis, from the theory of Riemann surfaces and Teichmüller and Moduli spaces to more recent developments such as holomorphic dynamics and three-dimensional hyperbolic geometry. We hope the developments presented in this book encourage new applications of quasiconformal mappings in these areas.

Preface. How to Use This Book. 1. Surfaces and Straightness. When Do You Call a Straight Line? How Do You Construct a Straight Line? Local (and Infinitesimal) Straightness. Intrinsic Straight Lines on Cylinders. Geodesics on Cones. Is Always Straight? Locally Isometric Surfaces. Local Coordinates for Cylinders and Cones. Geodesics in Local Coordinates. What Is Straight on a Sphere? Intrinsic Curvature on a Sphere. Local Coordinates on a Sphere. Strakes, Augers, and Helicoids. Surfaces of Revolution. Hyperbolic Plane. Surface as Graph of a Function z=f(x,y) 2. Extrinsic Curves. Introduction. Give Examples of F.O.V.'s. Archimedian Property. Vectors and Affine Linear Space. Smoothness and Tangent Directions. Curvature of a Curve in Space. Curvature of the Graph of a Function. Osculating Circle. Strakes. When a Curve Does Not Lie in a Plane. 3. Extrinsic Descriptions of Intrinsic Curvature. Smooth Surfaces and Tangent Planes. Extrinsic Curvature - Geodesics on Sphere. Intrinsic Curvature - Curves on Sphere. Intrinsic (Geodesic) Curvature. Geodesics on Surfaces--the Ribbon Test. Ruled Surfaces and the Converse of the Ribbon Test. 4. Tangent Space, Metric, Directional Derivative. The Tangent Space. Mean Value Theorem: Curves, Surfaces. Natural Parametrizations of Curves. Riemannian Metric. Riemannian Metric in Local Coordinates on a Sphere. Riemannian Metric in Local Coordinates on a Strake. Vectors in Extrinsic Local Coordinates. Measuring Using the Riemannian Metric. Directional Derivatives. Directional Derivative in Local Coordinates. Differentiating a Metric. Expressing Normal Curvature. Geodesic Local Coordinates. Differential Operator. Metric in Geodesic Coordinates. 5. Area, Parallel Transport, Intrinsic Curvature. The Area of a Triangle on a Sphere. Introducing Parallel Transport. The Holonomy of a Small Geodesic Triangle. Dissection of Polygons into Triangles. Gauss-Bonnet for Polygons on a Sphere. Parallel Fields and Intrinsic Curvature. Holonomy on Surfaces. Holonomy Explains Foucault's Pendulum. Intrinsic Curvature of a Surface. 6. Gaussian Curvature Extrinsically Defined. Pep Talk to the Reader. Gaussian Curvature, Extrinsic Definition. Second Fundamental Form. The Gauss Map. Gauss-Bonnet and Intrinsic Curvature. Matrix of the Second Fundamental. Mean Curvature and Minimal Surfaces. Celebration of Our Hard Work. 7. Applications of Gaussian Curvature. Gaussian Curvature in Local Coordinates. Curvature on Sphere, Strake, Catenoid. Circles, Polar Coordinates, and Curvature. Exponential Map and Shortest Is Straight. Ruled Surfaces and Ribbons. Surfaces with Constant Curvature. Curvature of the Hyperbolic Plane. 8. Intrinsic Local Descriptions and Manifolds. Covariant Derivative and Connections. Manifolds--Intrinsic and Extrinsic. Christoffel Symbols, Intrinsic Descriptions. Intrinsic Curvature and Geodesics. Lie Brackets, Coordinate Vector Fields. Riemann Curvature Tensors. Calculation of Curvature Tensors in Local Coordinates. Intrinsic Calculations in Examples. Appendix A. Linear Algebra--a Geometric Point of View. Where Do We Start? Geometric Affine Spaces. Vector Spaces. Inner Product--Lengths and Angles. Linear Transformations and Operators. Areas, Cross Products, and Triple Products. Volumes, Orientation, and Determinants. Eigenvalues and Eigenvectors. Introduction to Tensors. Appendix B. Analysis from a Geometric Point of View. Smooth Functions. Invariance of Domain. Inverse Function Theorem. Implicit Function Theorem. Appendix C. Computer Scripts. Standard Functions. Strake. Surfaces of Revolution. Surfaces as Graph of a Function. Tangent Vectors to Curves. Curvature and Tangent Vectors. Osculating Planes. Osculating Circles. Frenet Frame. Tangent Planes to Surfaces. Curves on a Surface. Extrinisic Curvature Vectors. The Three Curvature Vectors. Ruled Surfaces. Non-dissectable Polyhedron. Sign of (Gaussian) Curvature. Mulitple Principle Directions. Gauss Map. Helicoid to Catenoid. Bibliography. Notation Index. Subject Index.

The problem of prescribing Gaussian curvature on Riemann surface with conical singularity is considered. Let $(\Sigma,\beta)$ be a closed Riemann surface with a divisor $\beta$, and $K_\lambda=K+\lambda$, where $K:\Sigma\rightarrow\mathbb{R}$ is a Holder continuous function satisfying $\max_\Sigma K= 0$, $K\not\equiv 0$, and $\lambda\in\mathbb{R}$. If the Euler characteristic $\chi(\Sigma,\beta)$ is negative, then by a variational method, it is proved that there exists a constant $\lambda^\ast>0$ such that for any $\lambda\leq 0$, there is a unique conformal metric with the Gaussian curvature $K_\lambda$; for any $\lambda$, $0 \lambda^\ast$, there is no certain conformal metric having $K_{\lambda}$ its Gaussian curvature. This result is an analog of that of Ding and Liu \cite{Ding-Liu}, partly resembles that of Borer, Galimberti and Struwe \cite{B-G-Stru}, and generalizes that of Troyanov \cite{Troyanov} in the negative case.

Motivated by a recent work of Chen and Zhu (Commun Math Stat 1:369–385. 2013), we establish a Trudinger–Moser inequality on a compact Riemannian surface without boundary. The proof is based on blow-up analysis together with Carleson–Chang’s result (Bull Sci Math 110:113–127. 1986). This inequality is different from the classical one, which is due to Fontana (Comment Math Helv 68:415–454. 1993), since the Gaussian curvature is involved. As an application, we improve Chen–Zhu’s result as follows: a modified Liouville energy of conformal Riemannian metric has a uniform lower bound, provided that the Euler characteristic is nonzero and the volume of the conformal surface has a uniform positive lower bound.