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We study the solvability of one hypersingular integral equation on a smooth surface with boundary under the assumption that there exist global isothermal coordinates on the surface. The integral is understood in the sense of the Hadamard finite value, and the solution is sought in the class of functions that are Holder continuous, vanish at the boundary of the surface, and have surface gradient Holder continuous in some neighborhood of each point that does not lie on the boundary of the surface. The Fredholm alternative is proved for this equation. It is also shown that the results obtained can be applied to the boundary integral equation arising in the boundary value problem for the Helmholtz equation in the domain outside this surface with the Neumann condition on the surface.

The minimal surface (see [1]) is determined using the Weierstrass representation in three-dimensional space. The solution of the Dirac equation [2] in terms of spinors coincides with the representations of this surface with conservation of isothermal coordinates. The equation represented through the Dirac operator, which is included in the Manakov’s L, A, B triple [3] as equivalent to the modified Veselov-Novikov equation (mVN) [4]. The potential 𝑈 of the Dirac operator is the potential of representing a minimal surface. New solutions of the mVN equation are constructed using the pre-known potentials of the Dirac operator and this algorithm is said to be Moutard transformations [5]. Firstly, the geometric meaning of these transformations which found in [6], [7], gives us the definition of the inversion of the minimal surface, further after finding the exact solutions of the mVN equation, we can represent the inverted surfaces. And these representations of the new potential determine the soliton deformation [8], [9]. In 2014, blowing-up solutions to the mVN equation were obtained using a rigid translation of the initial Enneper surface in [6]. Further results were obtained for the second-order Enneper surface [10]. Now the soliton deformation of an inverted catenoid is found by smooth translation along the second coordinate axis. In this paper, in order to determine catenoid inversions, it is proposed to find holomorphic objects as Gauss maps and height differential [11]; the soliton deformation of the inverted catenoid is obtained; particular solution of modified Karteweg-de Vries (KdV) equation is found that give some representation of KdV surface [12],[13].

Kao što i sam naslov diplomskog rada kaže, minimalne plohe ključni su pojam ovog rada. Taj pojam ne obrađuje se temeljito u sklopu većine matematičkih programa diljem Hrvatske, ali itekako ima svoju važnost kroz povijest matematike. U ovom diplomskom radu, najprije smo uveli osnovne pojmove iz teorije krivulja i ploha, a zatim smo se posvetili minimalnim plohama i njihovim svojstvima, tj. pokazali smo da su minimalne plohe (one čija je srednja zakrivljenost jednaka nuli) upravo plohe minimalne površine. Također, uveli smo i pojam izotermnih koordinata te smo izveli Weierstrass–Enneperovu reprezentaciju minimalnih ploha. Navedena reprezentacija nam omogućuje generiranje minimalnih ploha iz holomorfnih funkcija. Na kraju smo dali pregled nekih poznatijih minimalnih ploha, a najveću pažnju smo posvetili helikoidu i katenoidu. As we can see in the title, the key word of this thesis is minimal surfaces. That term is not too thoroughly studied in the majority of mathematics programs in Croatia but it has been very important throughout the history of mathematics. In the first part of this thesis, we recalled most important definitions and results from the theory of curves and surfaces and then we pointed out important properties regarding minimal surfaces. In another words, it was proven that minimal surfaces (mean curvature is equal to zero) are surfaces with the least area. We also introduced isothermal coordinates into our thesis after which we derived Weierstras–Enneper representation of minimal surfaces. Mentioned representation allows us to generate minimal surfaces from holomorphic functions. In the end, we gave an overview of some well-known minimal surfaces with emphasis on the two most famous minimal surfaces, helicoid and catenoid.

RESUMEN: Las superficies minimales cobran un especial interés a lo largo de la historia, es por eso que estudiaremos distintas propiedades para identificar este tipo de superficies. Conoceremos los distintos avances a lo largo de la historia en la investigación de estas superficies. A su vez resultará interesante la importante conexión de las superficies minimales con el análisis complejo, nos centraremos en el estudio de coordenadas isotermas y la relación que mantienen con este tipo de superficies. Como último punto a destacar, veremos que otro de los objetos principales de estudio en este trabajo será la representación de Weierstrass y su importante relación con las superficies minimales, para lo que analizaremos concretamente el ejemplo de la superficie minimal de Costa. ABSTRACT: Minimal surfaces have been of special interest throughout history, that is why we will study different properties to identify this type of surfaces. We will learn about the different advances throughout history in the investigation of these surfaces. At the same time it will be interesting the important connection of the minimal surfaces with the complex analysis, we will focus on the study of isothermal coordinates and the relation that they maintain with this type of surfaces. As last point to emphasize, we will see that another of the main objects of study in this work will be the Weierstrass representation and its important relation with the minimal surfaces, for which we will analyze concretely the example of the minimal surface of Costa. Grado en Matemáticas

The dynamics of low energy electrons in general static strained graphene surface is modelled mathematically by the Dirac equation in curved space-time. In Cartesian coordinates, a parametrization of the surface can be straightforwardly obtained, but the resulting Dirac equation is intricate for general surface deformations. Two different strategies are introduced to simplify this problem: the diagonal metric approximation and the change of variables to isothermal coordinates. These coordinates are obtained from quasi-conformal transformations characterized by the Beltrami equation, whose solution gives the mapping between both coordinate systems. To implement this second strategy, a least square finite-element numerical scheme is introduced to solve the Beltrami equation. The Dirac equation is then solved via an accurate pseudo-spectral numerical method in the pseudo-Hermitian representation that is endowed with explicit unitary evolution and conservation of the norm. The two approaches are compared and applied to the scattering of electrons on Gaussian shaped graphene surface deformations. It is demonstrated that electron wave packets can be focused by these local strained regions., 19 pages, 8 figures

Lee’s exact method was developed to enable the Gauss–Kruger (GK) projection to be implemented without iterative procedures via the expansion of the intermediate mapping (Thompson projection) into series approximations in terms of isothermal coordinates $$\left( \psi , \lambda \right) $$ for the forward mapping and GK coordinates $$\left( x, y \right) $$ for the reverse mapping. The straightforward procedures expressed by new formulas for both forward and reverse mapping of the GK projection were composed by three sequential steps which essentially reveal the mapping procedures and intrinsic properties of the GK projection: The first step of deriving the isothermal coordinates $$\left( \psi , \lambda \right) $$ from the geodetic coordinates $$\left( \varphi , \lambda \right) $$ specifies a conformal mapping (i.e., the Normal Mercator projection) of the Earth ellipsoid surface into the Euclidean plane excluding the South and North poles, and the subsequent two steps allow the GK projection to be expressed analytically via the elliptic functions and integrals. Based on the three-step procedure, the conformality and singularities over the entire ellipsoid of the Normal Mercator, Thompson and GK projections were analyzed and the fundamental domains of them were determined. With respect to the precision and efficiency, it was verified that the new algorithm and the complex latitude method had equivalent precision levels for the same orders of the third flatting n with the Kruger-n series from $$n^2$$ to $$n^{12}$$ but slower about 0.19 to $${0.21}\,\upmu \hbox {s}$$ than the Kruger-n series for a GK coordinates calculating. However, the new formulas provide series approximations for the forward mapping of the Thompson projection and projective transformations for the Normal Mercator, Thompson and GK projections.

International audience; We report on Lavrentieff's paper ``Sur une classe de repr\'esentations continues'' (On a class of continuous representations), published in 1935, in which he introduced quasiconformal mappings under the name ``almost analytic functions." He set the bases of this theory, obtaining several results that show that these functions constitute natural generalizations of conformal mappings. The results include an existence and uniqueness theorem which constitutes an early version of the measurable Riemann mapping theorem, a result on the existence of isothermal coordinates, a generalization of Picard's big theorem to the setting of almost analytic functions, and several results on the type problem.

Using the method of blow-up analysis, we obtain two sharp Trudinger-Moser inequalities on a compact Riemann surface with smooth boundary, as well as the existence of the corresponding extremals. This generalizes early results of Chang-Yang [7] and the first named author [32], and complements Fontana's inequality of two dimensions [15]. The blow-up analysis in the current paper is far more elaborate than that of [32], and particularly clarifies several ambiguous points there. In precise, we prove the existence of isothermal coordinate systems near the boundary, the existence and uniform estimates of the Green function with the Neumann boundary condition. Also our analysis can be applied to the Kazdan-Warner problem and the Chern-Simons Higgs problem on compact Riemman surfaces with smooth boundaries., Comment: 37 pages

International audience; This paper introduces a novel algorithm for the online estimate of the Riemannian mixture model parameters. This new approach counts on Riemannian geometry concepts to extend the well-known Tit-terington approach for the online estimate of mixture model parameters in the Euclidean case to the Riemannian manifolds. Here, Riemannian mixtures in the Riemannian manifold of Symmetric Positive Definite (SPD) matrices are analyzed in details, even if the method is well suited for other manifolds.

Lorentz Geometrisi'nin özellikle genel görelilik ve matematiksel kozmolojiyi içerenüzere, bir çok farklı, gelişen araştırma alanlarında yararlı olduğu görülmüştür.Paralel dejenere dağılımı karakterize kabul eden Walker manifoldları, Lorentzmanifold yapısından türemiştir.Bu tezde, üç boyutlu Walker manifoldlar üzerindeki minimal yüzeyler üzerineçalıştık ve bu yapılar üzerindeki minimal yüzeyler için denklemler türettik.Özellikle, düzgün bir fonksiyon grafiği tarafından temsil edilen yüzeyler üzerinde çalıştık. Çalışmamız, bu denklemleri türetmek için kolay bir yöntem sağlayanLorentz izotermal (eşışıl) koordinatlarla yakından ilgilidir ve bu koordinatlar manifoldüzerindeki her yüzey için bölgesel olarak tanımlanmıştır. Minimal yüzeylerdeortalama eğriliğin sıfır olduğu gerçeğini ve seçilen koordinatlarla konumlanmış geometriksınırlamayı kullanarak, karşılık gelen fonksiyonun belli şartlardaki minimalfonksiyon grafik sınıflaırını elde ettik. Lorentzian Geometry has shown to be very useful in a wide range of studiesincluding many diverse research fields, especially in the theory of general relativityand mathematical cosmology. A Walker manifold descends from the structure ofLorentzian manifolds which is characterized by admitting a parallel degeneratedistribution.In the present thesis, we investigate and derive the equations of minimal surfaceson three-dimensional Walker manifolds, with a particular interest on thosesurfaces which are represented by the graph of a smooth function. Our study isclosely related with (Lorentzian) isothermal coordinates which provide an easierapproach for deriving such equations, and they are locally defined for any surfaceon the underlying manifold. By using the well-known property of vanishing meancurvature for minimal surfaces, together with the geometric restrictions posed bythe chosen coordinates, we obtain a class of graphs of functions which are minimalunder certain conditions on the corresponding function. 41

In this paper, we are concerned with the relativistic string equations in de Sitter spacetime. Since the background spacetime is nonflat, the governing system of equations is semilinear under the usual isothermal coordinates, contrary to the linear wave equations in the Minkowski spacetime. Based on the null condition enjoyed by the system and the method of [Luli et al., Adv. Math 329, 174–188 (2018)], we succeeded in proving the stability of the traveling wave solution to the relativistic string equations. We note that the traveling wave solution should be small in the sense of weighted energy.

We consider a region $M$ in $\mathbb{R}^n$ with boundary $\partial M$ and a metric $g$ on $M$ conformal to the Euclidean metric. We analyze the inverse problem, originally formulated by Dix, of reconstructing $g$ from boundary measurements associated with the single scattering of seismic waves in this region. In our formulation the measurements determine the shape operator of wavefronts outside of $M$ originating at diffraction points within $M$. We develop an explicit reconstruction procedure which consists of two steps. In the first step we reconstruct the directional curvatures and the metric in what are essentially Riemmanian normal coordinates; in the second step we develop a conversion to Cartesian coordinates. We admit the presence of conjugate points. In dimension $n \geq 3$ both steps involve the solution of a system of ordinary differential equations. In dimension $n=2$ the same is true for the first step, but the second step requires the solution of a Cauchy problem for an elliptic operator which is unstable in general. The first step of the procedure applies for general metrics.

The problem of explicit form of the metric of revolution on Bertrand’s Riemannian manifolds in particular coordinates is solved. Connections with earlier results of M. Santoprete are discussed.

In this paper we introduce a notion of scattering theory for the Laplace-Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another condition is certain bounds of derivatives up to order one of the trace of this quantity. These conditions are shown to be optimal for existence and completeness of a wave operator. Our theory does not involve prescribed asymptotic behaviour of the metric at infinity (like asymptotic Euclidean or hyperbolic metrics studied previously in the literature). A consequence of the theory is spectral theory for the Laplace-Beltrami operator including identification of the continuous spectrum and absence of singular continuous spectrum.

The problem of nonlinear alignment of functions is both fundamental and extremely important in pattern recognition. Most common approaches for alignment, including dynamic time warping (DTW), use penalized-\(\mathbb {L}^2\) minimization that has significant shortcomings, including asymmetry. A recent mathematical framework, based on an elastic Riemannian metric and square-root velocity functions, overcomes these shortcomings. The time warping problem is currently solved using a dynamic programming algorithm (DPA) which relies heavily on dense sampling of functions and can be computationally expensive. Here we present a novel theory (and algorithms) for finding the exact pairwise alignment between functions, which uses an efficient sampling of functions, restricted to their change points. In many cases, the computational cost for matching is reduced by orders of magnitude. We demonstrate the superiority of this method over the DPA using several simulated and real datasets.

Riemannian optimization has been widely used to deal with the fixed low-rank matrix completion problem, and Riemannian metric is a crucial factor of obtaining the search direction in Riemannian optimization. This paper proposes a new Riemannian metric via simultaneously considering the Riemannian geometry structure and the scaling information, which is smoothly varying and invariant along the equivalence class. The proposed metric can make a tradeoff between the Riemannian geometry structure and the scaling information effectively. Essentially, it can be viewed as a generalization of some existing metrics. Based on the proposed Riemanian metric, we also design a Riemannian nonlinear conjugate gradient algorithm, which can efficiently solve the fixed low-rank matrix completion problem. By experimenting on the fixed low-rank matrix completion, collaborative filtering, and image and video recovery, it illustrates that the proposed method is superior to the state-of-the-art methods on the convergence efficiency and the numerical performance.

The dynamics of a membrane is a coupled system comprising a moving elastic surface and an incompressible membrane fluid. We will consider a reduced elastic surface model, which involves the evolution equations of the moving surface, the dynamic equations of the two-dimensional fluid, and the incompressible equation, all of which operate within a curved geometry. In this paper, we prove the local existence and uniqueness of the solution to the reduced elastic surface model by reformulating the model into a new system in the isothermal coordinates. One major difficulty is that of constructing an appropriate iterative scheme such that the limit system is consistent with the original system., The introduction is rewritten

The Laplace-Beltrami operator of a smooth Riemannian manifold is determined by the Riemannian metric. Conversely, the heat kernel constructed from the eigenvalues and eigenfunctions of the Laplace-Beltrami operator determines the Riemannian metric. This work proves the analogy on Euclidean polyhedral surfaces (triangle meshes), that the discrete heat kernel and the discrete Riemannian metric (unique up to a scaling) are mutually determined by each other. Given a Euclidean polyhedral surface, its Riemannian metric is represented as edge lengths, satisfying triangle inequalities on all faces. The Laplace-Beltrami operator is formulated using the cotangent formula, where the edge weight is defined as the sum of the cotangent of angles against the edge. We prove that the edge lengths can be determined by the edge weights unique up to a scaling using the variational approach. The constructive proof leads to a computational algorithm that finds the unique metric on a triangle mesh from a discrete Laplace-Beltrami operator matrix.

We find a one-parameter family of Riemannian metrics on for for some number with the following property: is the Euclidean metric on , the scalar curvatures of are strictly decreasing in t in the open unit ball and is isometric to the Euclidean metric in the complement of the ball.

The conformal deformations are contained in two classes of mappings quasiconformal and harmonic mappings. In this paper we consider the intersection of these classes. We show that, every K quasiconformal harmonic mapping between \({{\fancyscript{C}}^{1,\mu} (0 < \mu\le 1)}\) surfaces with \({{\fancyscript{C}}^{1,\mu}}\) boundary is a Lipschitz mapping. This extends some recent results of several authors where the same problem has been considered for plane domains. As an application it is given an explicit Lipschitz constant of normalized isothermal coordinates of a disk-type minimal surface in terms of boundary curve only. It seems that this kind of estimates are new for conformal mappings of the unit disk onto a Jordan domain as well.

Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ -convergence under the proper scaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a W 2,2 isometric immersion of a given 2 d metric into .

We continue the study of minimal singular surfaces obtained by a minimization of a weighted energy functional in the spirit of J. Douglas’s approach to the Plateau problem. Modeling soap films spanning wire frames, a singular surface is the union of three disk-type surfaces meeting along a curve which we call the free boundary. We obtain an a priori regularity result concerning the real analyticity of the free boundary curve. Using the free boundary regularity of the harmonic map, we construct local harmonic isothermal coordinates for the minimal singular surface in a neighborhood of a point on the free boundary. Applications of the local uniformization are discussed in relation to H. Lewy’s real analytic extension of minimal surfaces.

We mainly investigate some properties of quasiconformal mappings between smooth 2-dimensional surfaces with boundary in the Euclidean space, satisfying certain partial differential equations (inequalities) concerning Laplacian, and in particular satisfying Laplace equation and show that these mappings are Lipschitz. Conformal parametrization of such surfaces and the method developed in our paper (Kalaj and Mateljevic, J Anal Math 100:117–132, 2006) have important role in this paper.

We develop a method for reconstructing the conformal factor of a Riemannian metric and the magnetic field on a surface from the scattering relation associated to the corresponding magnetic flow. The scattering relation maps a starting point and direction of a magnetic geodesic into its end point and direction. The key point in the reconstruction is the interplay between the magnetic ray transform, the fiberwise Hilbert transform on the circle bundle of the surface, and the Laplace-Beltrami operator of the underlying Riemannian metric.

A Riemannian metric g on a compact boundaryless manifold is said to be locally audible if the following statement is true for every metric g′ sufficiently close to g: if g and g′ are isospectral then they are isometric. The local audibility is proved of a metric of constant negative sectional curvature.

The real homology of a compact Riemannian manifold M is naturally endowed with the stable norm. The stable norm on H-1 (M. R) arises from the Riemannian length functional by homogenization. It is difficult and interesting to decide which norms on the finite-dimensional vector space H-1 (M,R) are stable norms of a Riemannian metric on M. If the dimension of M is at least three, I. Babenko and F. Balacheff proved in [I. Babenko, F Balacheff, Sur la forme de la boule unite de la norme stable uniclimensionnelle, Manuscripta Math. 119 (3) (2006) 347-358] that every polyhedral norm ball in H I (M, R), whose vertices are rational with respect to the lattice of integer classes in HI(M,R), is the stable norm ball of a Riemannian metric on M. This metric can even be chosen to be conformally equivalent to any given metric. In [I. Babenko, F. Balacheff, Sur la forme de la boule unite de la norme stable uniclimensionnelle, Manuscripta Math. 119 (3) (2006) 347-358], the stable norm induced by the constructed metric is computed by comparing the metric with a polyhedral one. Here we present an alternative construction for the metric. which remains in the geometric framework of smooth Riemannian metrics. (C) 2009 Elsevier B.V. All rights reserved.

We show with a new variational approach that any Riemannian metric on a multiply connected schlicht domain in ℝ2 can be represented by globally conformal parameters. This yields a “Riemannian version” of Koebe's mapping theorem.

In two previous articles [Phys. Rev. D71 (2005) 124307 (gr-qc/0503007), and gr-qc/0607001] we have discussed several "algorithmic" techniques that permit one (in a purely mechanical way) to generate large classes of general relativistic static perfect fluid spheres. Working in Schwarzschild curvature coordinates, we used these algorithmic ideas to prove several "solution-generating theorems" of varying levels of complexity. In the present article we consider the situation in other coordinate systems: In particular, in general diagonal coordinates we shall generalize our previous theorems, in isotropic coordinates we shall encounter a variant of the so-called "Buchdahl transformation", while in other coordinate systems (such as Gaussian polar coordinates, Synge isothermal coordinates, and Buchdahl coordinates) we shall find a number of more complex "Buchdahl-like transformations" and "solution-generating theorems" that may be used to investigate and classify the general relativistic static perfect fluid sphere. Finally by returning to general diagonal coordinates and making a suitable ansatz for the functional form of the metric components we place the Buchdahl transformation in its most general possible setting., Comment: 23 pages

With the advance of human connectome research, there are great interests in computing diffeomorphic maps of brain surfaces with rich connectivity features. In this paper, we propose a novel framework for connectivity-driven surface mapping based on Riemannian metric optimization on surfaces (RMOS) in the Laplace-Beltrami (LB) embedding space. The mathematical foundation of our method is that we can use the pullback metric to define an isometry between surfaces for an arbitrary diffeomorphism, which in turn results in identical LB embeddings from the two surfaces. For connectivity-driven surface mapping, our goal is to compute a diffeomorphism that can match a set of connectivity features defined over anatomical surfaces. The proposed RMOS approach achieves this goal by iteratively optimizing the Riemannian metric on surfaces to match the connectivity features in the LB embedding space. At the core of our framework is an optimization approach that converts the cost function of connectivity features into a distance measure in the LB embedding space, and optimizes it using gradients of the LB eigen-system with respect to the Riemannian metric. We demonstrate our method on the mapping of thalamic surfaces according to connectivity to ten cortical regions, which we compute with the multi-shell diffusion imaging data from the Human Connectome Project (HCP). Comparisons with a state-of-the-art method show that the RMOS method can more effectively match anatomical features and detect thalamic atrophy due to normal aging.

We formulate some conditions for the normal and compact solvability of the operator of exterior derivation on the cylindrical manifolds equipped with some Riemannian metrics. Some analogous results were obtained in the particular case of warped cylinders [1].

Some results on existence of global Chebyshev coordinates on a Riemannian manifold or, more generally, on Aleksandrov surface are proved. For instance, if the positive and the negative parts of integral curvature of a Riemannian manifold M are less than 2\pi each, then there exist global Chebyshev coordinates on M. These conditions are optimal. Such coordinates help to get bi-Lipschitz maps between surfaces.}, Comment: This is a corrected version of our previous submisson

We present a new variational proof of the well-known fact that every Riemannian metric on a two-dimensional, simply connected domain with boundary can be represented by globally conformal parameters. From this the corresponding result for a metric on S2 is derived.

A new method to rectify monoscopic digital images and generate orthoimages of the Earth's surface is described. It replaces the standard procedure, which transfers the perspective projection of a frame photograph to an orthographic projection of pixels onto a reference plane using corresponding corrections. Instead, the perspective forward projection is kept but every pixel is vertically mapped along the surface normal onto a curved reference surface, for example, the ellipsoid of the World Geodetic System 1984 under the condition that a precise enough surface elevation model is available. The gained ellipsoidal coordinates (latitude, longitude and height) of each pixel are then transformed into isothermal coordinates like the Universal Transverse Mercator coordinates. Their differential geometric characteristics allow mapping every pixel to a reference plane producing, after some interpolation between irregularly spaced pixels, a photomap with the same geometric properties as any other topographic map. The suitability of the method is demonstrated by two photomaps from Ankara, Turkey, which are compared to high-quality topographic maps whereby the average position errors are about 2-3 pixels.

The Gaussian curvature of a two-dimensional Riemannian manifold is uniquely determined by the choice of the metric. The formulas for computing the curvature in terms of components of the metric, in isothermal coordinates, involve the Laplacian operator and therefore, the problem of finding a Riemannian metric for a given curvature form may be viewed as a potential theory problem. This problem has, generally speaking, a multitude of solutions. To specify the solution uniquely, we ask that the metric have the mean value property for harmonic functions with respect to some given point. This means that we assume that the surface is simply connected and that it has a smooth boundary. In terms of the so-called metric potential, we are looking for a unique smooth solution to a nonlinear fourth order elliptic partial differential equation with second order Cauchy data given on the boundary. We find a simple condition on the curvature form which ensures that there exists a smooth mean value surface solution. It reads: the curvature form plus half the curvature form for the hyperbolic plane (with the same coordinates) should be $\le0$. The same analysis leads to results on the question of whether the canonical divisors in weighted Bergman spaces over the unit disk have extraneous zeros. Numerical work suggests that the above condition on the curvature form is essentially sharp. Our problem is in spirit analogous to the classical Minkowski problem, where the sphere supplies the chart coordinates via the Gauss map., Comment: 32 pages

In this paper, we define and study the Weil–Petersson geometry. Under the framework of the Weil–Petersson geometry, we study the Weil–Petersson metric and the Hodge metric. Among the other results, we represent the Hodge metric in terms of the Weil–Petersson metric and the Ricci curvature of the Weil–Petersson metric for Calabi–Yau fourfold moduli. We also prove that the Hodge volume of the moduli space is finite. Finally, we proved that the curvature of the Hodge metric is bounded if the Hodge metric is complete and the dimension of the moduli space is 1.

We consider local finite energy coordinates associated with a strongly local regular Dirichlet form on a metric measure space. We give coordinate formulas for substitutes of tangent spaces, for gradient and divergence operators and for the infinitesimal generator. As examples we discuss Euclidean spaces, Riemannian local charts, domains on the Heisenberg group and the measurable Riemannian geometry on the Sierpinski gasket.