Search

We investigate harmonic maps on almost contact metric manifolds which are locally conformal to almost cosymplectic manifolds. We obtain the necessary and sufficient conditions for the holomorphy to imply harmonicity and then we find obstructions to the existence of non-constant pluriharmonic maps. We also establish some results on the stability of the identity map on a locally conformal almost cosymplectic manifold of pointwise constant φ-holomorphic sectional curvature. [ABSTRACT FROM AUTHOR]

We study the strict convexity of the energy function of harmonic maps at their critical points from a Riemann surface to a Riemann surface, or to the product of negatively curved surfaces. When the target is a Riemann surface and when the map is of nonzero degree, we obtain a precise formula for the second derivative of the energy function along a Weil–Petersson geodesic, which implies that the energy function is strictly convex at its critical points. When the target is the product of two surfaces where each projection of the harmonic map is homotopic to a covering map, we also prove the strict convexity of the associated energy function. As an application we prove that the energy function has a unique critical point in these cases. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce Clairaut conformal submersions between Riemannian manifolds. First, we find necessary and sufficient conditions for conformal submersions to be Clairaut conformal submersions. In particular, we obtain Clairaut relation for geodesics on the total manifolds of conformal submersions, and prove that Clairaut conformal submersions have constant dilation along their fibers, which are totally umbilical, with mean curvature being gradient of a function. Further, we calculate the scalar and Ricci curvatures of the vertical distributions of the total manifolds. Moreover, we find a necessary and sufficient condition for Clairaut conformal submersions to be harmonic. For a Clairaut conformal submersion we find conformal changes of the metric on its domain or image, that give a Clairaut Riemannian submersion, a Clairaut conformal submersion with totally geodesic fibers, or a harmonic Clairaut submersion. Finally, we give two non-trivial examples of Clairaut conformal submersions to illustrate the theory and present a local model of every Clairaut conformal submersion with integrable horizontal distribution. [ABSTRACT FROM AUTHOR]

The paper deals with the twisted Sasaki metric on the unit tangent bundle of n-dimensional Riemannian manifold Mn. The main purpose of the research is to find deformations that preserve the existence harmonic left-invariant unit vector fields on 3-dimensional unimodular Lie groups G with the left invariant metric and harmonic maps G → T1G in case of twisted Sasaki metric on the unit tangent bundle. The necessary and sufficient conditions for harmonicity of left-invariant unit vector field and map Mn → T1Mn are obtained. The necessary and sufficient conditions for harmonicity of left-invariant unit vector field and map M² → T1M² with respect to some orthonormal frame are obtained. Left-invariant harmonic unit vector fields and harmonic maps G → T1G, where G is a three-dimensional unimodular Lie group with left-invariant metric, using some orthonormal frame are described. Left-invariant harmonic unit vector fields which determine harmonic maps G → T1G, where G is a three-dimensional unimodular Lie group with left-invariant metric in the particular case of twisted Sasaki metric, namely the vertical rescaled metric are classified. [ABSTRACT FROM AUTHOR]

In this paper, we introduce Riemannian warped product map as a generalization of warped product isometric immersion and Riemannian warped product submersion with examples and obtain some characterizations. First, we define Riemannian warped product map and find conditions for a Riemannian map to be Riemannian warped product map. We show that Riemannian warped product map is a composition of Riemannian warped product submersion followed by warped product isometric immersion locally. In addition, we show that Riemannian warped product map satisfies the generalized eikonal equation which is a well known bridge between geometrical and physical optics. We also find necessary and sufficient conditions for the fibers, range space of the derivative map of Riemannian warped product map and horizontal distributions to be totally geodesic and minimal. Further, we give some fundamental geometric properties for the study of such smooth maps. Precisely, we construct Gauss formula (second fundamental form), Weingarten formula and tension field. We obtain necessary and sufficient conditions for a Riemannian warped product map to be totally geodesic, harmonic and umbilical. Comparatively, we analyse the obtained results with the existing results for a Riemannian map between Riemannian manifolds. [ABSTRACT FROM AUTHOR]

In this paper, a new method is developed for tool path planning on triangular mesh surfaces with consideration of the scallop height restriction and the path smoothness. This method first maps the triangular mesh surface into a unit disk region by using a harmonic map algorithm, and then the shortest boundary path graph (SBPG) is constructed on the unit disk region to describe the shortest geodesic distance from each mesh vertex to the surface boundary. The tool path is then obtained by inversely mapping the contours of SBPG from the harmonic mapped region to the physical space of mesh surface. During this process, a subdivision method is used to boost the computing efficiency and a smoothing treatment is conducted on the SBPG to improve the path smoothness. The tool path planning is performed starting from the surface boundary in an iteration process. Taking the level difference of SBPG contours as the initial path interval and being supplemented by a correction process, the maximal step distance between any two paths, which meets the requirement of scallop height restriction, can be determined efficiently. Typical simulation cases and experiments are carried out to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The radial map u(x) = x ‖ x ‖ is a well-known example of a harmonic map from R m - { 0 } into the spheres S m - 1 with a point singularity at x = 0. In Nakauchi (Examples Counterexamples 3:100107, 2023), the author constructed recursively a family of harmonic maps u (n) into S m n - 1 with a point singularity at the origin (n = 1 , 2 , …) , such that u (1) is the above radial map. It is known that for m ≥ 3, the radial map u (1) is not only stable as a harmonic map but also a minimizer of the energy of harmonic maps. In this paper, we show that for n ≥ 2, u (n) may be unstable as a harmonic map. Indeed we prove that under the assumption n > 3 - 1 2 (m - 1) (m ≥ 3 , n ≥ 2) , the map u (n) is unstable as a harmonic map. It is remarkable that they are unstable and our result gives many examples of unstable harmonic maps into the spheres with a point singularity at the origin. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with harmonic maps defined on compact Riemannian manifolds and with values in homogenous strongly harmonic manifolds. We show some results on the Morse index by varying these maps along suitable conformal vector fields. [ABSTRACT FROM AUTHOR]

The classical Fatou theorem identifies bounded harmonic functions on the unit disk with bounded measurable functions on the boundary circle. We extend this theorem to bounded harmonic maps. [ABSTRACT FROM AUTHOR]

In this note, we study the Dirichlet problem for harmonic maps from strongly rectifiable spaces into regular balls in CAT (1) space. Under the setting, we prove that the Korevaar–Schoen energy admits a unique minimizer. [ABSTRACT FROM AUTHOR]

The radial map u(x)=x‖x‖is a well-known example of a harmonic map into the spheres with a point singularity at x=0. In our previous paper (Misawa and Nakauchi, 2022) we give two examples of harmonic maps into the standard spheres of higher dimension with a singularity of a polynomial of x1‖x‖,⋯xm‖x‖of degree 2 and degree 3 respectively. In Fujioka (2020) uses our arguments to give an example of a harmonic map into the sphere with a singularity of a polynomial of degree 4. In this paper we give a family of examples of harmonic maps with a point singularity of a polynomial of higher degree.

In this paper, we study conformal submersions from Ricci solitons to Riemannian manifolds with non-trivial examples. First, we study some properties of the O'Neill tensor A in the case of conformal submersion. We also find a necessary and sufficient condition for conformal submersion to be totally geodesic and calculate the Ricci tensor for the total manifold of such a map with different assumptions. Further, we consider a conformal submersion F : M → N from a Ricci soliton to a Riemannian manifold and obtain necessary conditions for the fibers of F and the base manifold N to be Ricci soliton, almost Ricci soliton and Einstein. Moreover, we find necessary conditions for a vector field and its horizontal lift to be conformal on N and (Ker F ∗) ⊥ , respectively. Also, we calculate the scalar curvature of Ricci soliton M. Finally, we obtain a necessary and sufficient condition for F to be harmonic. [ABSTRACT FROM AUTHOR]

In a previous paper, we proved that a quasi-isometric map f : X → Y between two pinched Hadamard manifolds X and Y is within bounded distance from a unique harmonic map. We extend this result to maps f : Γ \ X → Y , where Γ is a convex cocompact discrete group of isometries of X and f is locally quasi-isometric at infinity. [ABSTRACT FROM AUTHOR]

Let G k , n be the Grassmannian of oriented subspaces of dimension k of R n with its canonical Riemannian metric. We study the energy of maps assigning to each P ∈ G k , n a unit vector normal to P. They are sections of a sphere bundle E k , n 1 over G k , n . The octonionic double and triple cross products induce in a natural way such sections for k = 2 , n = 7 and k = 3 , n = 8 , respectively. We prove that they are harmonic maps into E k , n 1 endowed with the Sasaki metric. This, together with the well-known result that Hopf vector fields on odd dimensional spheres are harmonic maps into their unit tangent bundles, allows us to conclude that all unit normal sections of the Grassmannians associated with cross products are harmonic. In a second instance we analyze the energy of maps assigning an orthogonal complex structure J P on P ⊥ to each P ∈ G 2 , 8 . We prove that the one induced by the octonionic triple product is a harmonic map into a suitable sphere bundle over G 2 , 8 . This generalizes the harmonicity of the canonical almost complex structure of S 6 . [ABSTRACT FROM AUTHOR]

This paper is devoted to investigate harmonic functions on vector bundle manifolds endowed with the class of spherically symmetric metrics. We will study three classes of harmonic functions on total spaces of vector bundles. We attempt to establish the link between harmonic function theories on the base and total space. [ABSTRACT FROM AUTHOR]

Let Y = (Y , d) be a CAT (0) space which is either proper or of finite telescopic dimension, and Γ a countable group equipped with a symmetric and nondegenerate probability measure μ . Suppose that Γ acts on Y via a homomorphism ρ : Γ → Isom (Y) , where Isom (Y) denotes the isometry group of Y, and that the action given by ρ has finite second moment with respect to μ . We show that if ρ (Γ) does not fix a point in the boundary at infinity ∂ Y of Y and the rate of escape l ρ (Γ) = l ρ (Γ , μ) associated to an action given by ρ vanishes, then there exists a flat subspace in Y that is left invariant under the action of ρ (Γ) . Note that if the rate of escape does not vanish, then we know that there exists an equivariant map from the Poisson boundary of (Γ , μ) into the boundary at infinity of Y by a result of Karlsson and Margulis. The key ingredient of the proof is μ -harmonic functions on Γ and μ -harmonic maps from Γ into Y. We prove a result similar to the above for an isometric action of Γ on a locally finite-dimensional CAT (0) space. [ABSTRACT FROM AUTHOR]

In this paper, we give some results about the generalized D-homothetic deformation on the almost contact metric manifold and we study the harmonicity and the biharmonicity relative to this type of deformation. In terms of applications, we have constructed several examples of harmonic and biharmonic maps. [ABSTRACT FROM AUTHOR]

In this paper, we consider hyperbolic Ricci-Bourguignon flow on a compact Riemannian manifold M coupled with the harmonic map flow between M and a fixed manifold N. At the first, we prove the unique short-time existence to solution of this system. Then, we find the second variational of some geometric structure of M along this system such as, curvature tensors. In addition, for emphasize the importance of hyperbolic Ricci-Bourguignon flow, we give some examples of this flow on Riemannian manifolds.

The paper focused on the mean curvature and totally geodesic property of the Reeb vector field on (a, ß) trans-Sasakian manifold M of dimension (2n+1) as a submanifold in the unit tangent bundle T1M with Sasaki metric gS. We give an explicit formula for the norm of mean curvature vector of the submanifold (M) (T1M, gS). As a byproduct, for the Reeb vector field, we get some known results concerning its minimality, harmonicity and the property to define a harmonic map. We prove that on connected proper trans-Sasakian manifold the Reeb vector field does not give rise to totally geodesic submanifold in T1M. On a-Sasakian the Reeb vector field is totally geodesic only if a = 1. On ß -Kenmotsu manifold the Reeb vector field is totally geodesic if and only if ß = ß2(1+ß2) 1-ß2. If M is compact, then ß = 0. [ABSTRACT FROM AUTHOR]

In this paper, we study an extension of the CPE conjecture to manifolds M which support a structure relating curvature to the geometry of a smooth map φ : M → N . The resulting system, denoted by (φ -CPE), is natural from the variational viewpoint and describes stationary points for the integrated φ -scalar curvature functional restricted to metrics with unit volume and constant φ -scalar curvature. We prove both a rigidity statement for solutions to (φ -CPE) in a conformal class, and a gap theorem characterizing the round sphere among manifolds supporting (φ -CPE) with φ a harmonic map. [ABSTRACT FROM AUTHOR]

We consider a kind of generalized harmonic maps, namely, the VT-harmonic maps. We prove an existence theorem for the Dirichlet problem of VT-harmonic maps from compact manifolds with boundary. [ABSTRACT FROM AUTHOR]

In this paper, we prove some refined estimate in the neck region when a sequence of harmonic maps from surfaces blow up. The new estimate puts more restrictions to the bubble and the weak limit map than the touching required by the classical no-neck theorem. As an application, we prove an inequality about the nullity and index when blow-up occurs. [ABSTRACT FROM AUTHOR]

We study harmonic maps from a 3-manifold with boundary to S 1 and prove a special case of Gromov dihedral rigidity of three-dimensional cubes whose dihedral angles are π / 2 . Furthermore, we give some applications to mapping torus hyperbolic 3-manifolds. [ABSTRACT FROM AUTHOR]

In this manuscript, we obtain necessary and sufficient conditions for a spacelike curve with timelike first binormal vector field to be a bi-ƒ-harmonic curve in a 4-dimensional conformally flat, quasi-conformally flat and conformally symmetric Lorentzian para-Sasakian manifold. [ABSTRACT FROM AUTHOR]

Given a closed manifold of dimension at least three, with non-trivial homotopy group π 3 (M) and a generic metric, we prove that there is a finite collection of harmonic spheres with Morse index bounded by one, with sum of their energies realizing a geometric invariant width. [ABSTRACT FROM AUTHOR]

In this paper, we study Clairaut Riemannian maps whose total manifolds admit a Ricci soliton and give a nontrivial example of such Clairaut Riemannian maps. First, we calculate Ricci tensors and scalar curvature of total manifolds of Clairaut Riemannian maps. Then we obtain necessary conditions for the fibers of such Clairaut Riemannian maps to be Einstein and almost Ricci solitons. We also obtain a necessary condition for vector field α ̇ to be conformal, where α is a geodesic curve on total manifold of Clairaut Riemannian map. Further, we show that if total manifolds of Clairaut Riemannian maps admit a Ricci soliton with the potential mean curvature vector field of ker F ∗ then the total manifolds of Clairaut Riemannian maps also admit a gradient Ricci soliton and obtain a necessary and sufficient condition for such maps to be harmonic by solving Poisson equation. [ABSTRACT FROM AUTHOR]

We study the blow-up analysis and qualitative behavior for a sequence of harmonic maps with free boundary from degenerating bordered Riemann surfaces with uniformly bounded energy. With the help of Pohozaev type constants associated to harmonic maps defined on degenerating collars, including vertical boundary collars and horizontal boundary collars, we establish a generalized energy identity. [ABSTRACT FROM AUTHOR]

For a given minimal surface in the n-sphere, two ways to construct a minimal surface in the m-sphere are given. One way constructs a minimal immersion. The other way constructs a minimal immersion which may have branch points. The branch points occur exactly at each point where the original minimal surface is geodesic. If a minimal surface in the 3-sphere is given, then these ways construct Lawson's polar variety and bipolar surface. [ABSTRACT FROM AUTHOR]

We equip an arbitrary vector bundle over a Riemannian manifold, endowed with a fiber metric and a compatible connection, with a spherically symmetric metric (cf. [4]), and westudy harmonicity of its sections firstly as smooth maps and then as critical points of the energy functional with variations through smooth sections.We also characterize vertically harmonic sections. Finally, we give some examples of special vector bundles, recovering in some situations some classical harmonicity results. [ABSTRACT FROM AUTHOR]

The radial map w(x) = x ‖x‖−1 is a well-known example of harmonic maps and p-harmonic ones into spheres with a point singularity. In this paper we give two examples of harmonic maps and p-harmonic ones into spheres of higher dimensions with the singularity xixj ‖x‖−2 or the singularity xixjxk ‖x‖−3. [ABSTRACT FROM AUTHOR]

Given a compact connected Riemann surface Σ of genus g Σ ≥ 2 , and an effective divisor D = ∑ i n i x i on Σ with degree (D) < 2 (g Σ − 1) , there is a unique cone metric on Σ of constant negative curvature − 4 such that the cone angle at each point x i is 2 π n i [R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc.103 (1988) 222–224; M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821]. We describe the Higgs bundle on Σ corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on Σ parametrized by a nonempty open subset of H 0 (Σ , K Σ ⊗ 2 ⊗ Σ (− 2 D)) that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin's results in [N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126] for the case D = 0. [ABSTRACT FROM AUTHOR]

Let f be a harmonic map from a Riemann surface to a Riemannian n-manifold. We prove that if there is a holomorphic diffeomorphism h between open subsets of the surface such that f ∘ h = f , then f factors through a holomorphic map onto another Riemann surface. If such h is anti-holomorphic, we obtain an analogous statement. For minimal maps, this result is well known and is a consequence of the theory of branched immersions of surfaces due to Gulliver–Osserman–Royden. Our proof relies on various geometric properties of the Hopf differential. [ABSTRACT FROM AUTHOR]

We study the existence of harmonic maps and Dirac-harmonic maps from degenerating surfaces to a nonpositive curved manifold via the scheme of Sacks and Uhlenbeck. By choosing a suitable sequence of α -(Dirac-)harmonic maps from a sequence of suitable closed surfaces degenerating to a hyperbolic surface, we get the convergence and a cleaner energy identity under the uniformly bounded energy assumption. In this energy identity, there is no energy loss near the punctures. As an application, we obtain an existence result about (Dirac-)harmonic maps from degenerating (spin) surfaces. If the energies of the map parts also stay away from zero, which is a necessary condition, both the limiting harmonic map and Dirac-harmonic map are nontrivial. [ABSTRACT FROM AUTHOR]

We prove that a quasi-isometric map, and more generally a coarse embedding, between pinched Hadamard manifolds is within bounded distance of a unique harmonic map. [ABSTRACT FROM AUTHOR]

The 2-parameter family of certain homogeneous Lorentzian 3-manifolds, which includes Minkowski 3-space and anti-de Sitter 3-space, is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral representation formula for maximal spacelike surfaces in the homogeneous Lorentzian 3-manifolds is obtained. The normal Gauß map of maximal spacelike surfaces and its harmonicity are discussed. [ABSTRACT FROM AUTHOR]

We give an overview of the differential geometric and analytic aspects of Higgs bundles and their moduli spaces and highlight some of their interrelations with neighboring fields. We review various current developments in this subject and provide a discussion of a number of open problems. [ABSTRACT FROM AUTHOR]

The stability property of harmonic mappings plays a fundamental role in both mechanics and mathematical physics. This work is devoted to the investigation of the stability property on T -space forms, i.e. T -manifolds of constant φ -sectional curvature. According to Blair (1970) [8] , T -manifolds along with K -manifolds and S -manifolds represent the three important classes of manifolds with structural groups U (n) × O (s). Using the second variation formula for the identity function 1 M of a compact T -space form M , we first find the condition under which this function is stable. Thus, we establish that 1 M is stable, provided that the constant φ -sectional curvature of M is non-negative. Then, making use of the Weitzenböck formula, we prove that a (2 n + s) -dimensional compact T -space form of constant φ -sectional curvature c ≤ 0 is unstable, provided that the first eigenvalue of the Laplace-Beltrami operator is upper bounded by − c (n + 1). [ABSTRACT FROM AUTHOR]

Image deformation has ubiquitous usage in multimedia applications. It morphs one image into another through a seamless transition. Existing techniques either mainly focus on the correspondence mapping of interior features of the objects in two images, without considering object contours, or sketch contours manually, resulting in tedious work for users. Thus, we propose a 2D image deformation method, which extracts object contours automatically, considers both inner features and contours as constraints and preserves image features in terms of visual importance. Our method first automatically extracts the object contours in the source and target images and then allows users to sketch some interior features in both the images. Then, our method tessellates two images to generate two triangular meshes and builds a guaranteed bijective mesh mapping between them. We also prove the bijectivity of our mesh mapping and discuss its other desirable properties. Then, our method generates the intermediate images between the source and target images by calculating the intermediate meshes and pixels of each intermediate image. Our method realizes automatic contour extraction, provides an intuitive user interface and utilizes harmonic maps to establish a bijective mesh mapping. Therefore, it preserves more significant features with less distortion and works well for many image deformation cases in real time.

Let M be an open Riemann surface and let Λ ⊂ M be a closed discrete subset. In this paper, we prove the existence of complete conformal minimal immersions M → R n , n ≥ 3 , with prescribed values on Λ and whose generalized Gauss map M → CP n - 1 , n ≥ 3 , avoids n hyperplanes of CP n - 1 located in general position. In case n = 3 , we obtain complete nonflat conformal minimal immersions whose Gauss map M → S 2 omits two (antipodal) values of the sphere. This result is deduced as a consequence of an interpolation theorem for conformal minimal immersions M → R n into the Euclidean space R n , n ≥ 3 , with n - 2 prescribed components. [ABSTRACT FROM AUTHOR]

We consider harmonic maps u (z) : X z → N in a fixed homotopy class from Riemann surfaces X z of genus g ≥ 2 varying in the Teichmüller space T to a Riemannian manifold N with non-positive Hermitian sectional curvature. The energy function E (z) = E (u (z)) can be viewed as a function on T and we study its first and the second variations. We prove that the reciprocal energy function E (z) − 1 is plurisuperharmonic on Teichmüller space. We also obtain the (strict) plurisubharmonicity of log ⁡ E (z) and E (z). As an application, we get the following relationship between the second variation of logarithmic energy function and the Weil-Petersson metric if the harmonic map u (z) is holomorphic or anti-holomorphic and totally geodesic, i.e., (0.1) − 1 ∂ ∂ ¯ log ⁡ E (z) = ω W P 2 π (g − 1). We consider also the energy function E (z) associated to the harmonic maps from a fixed compact Kähler manifold M to Riemann surfaces { X z } z ∈ T in a fixed homotopy class. If u (z) is holomorphic or anti-holomorphic, then (0.1) is also proved. [ABSTRACT FROM AUTHOR]

In this paper, we characterize a class of biharmonic maps from and between doubly product manifolds in terms of theie warping function. Examples are constructed when all of the factors are Euclidean spaces. [ABSTRACT FROM AUTHOR]

In the maintenance work of tunnels, images are often used to detect diseases, but collections of tunnel images are limited by the tunnel environment and working time. Three-dimensional laser scanning technology can acquire high-precision tunnel information efficiently, and the main problem to be solved by using this technology to collect tunnel inner wall images is the dimensionality reduction of the laser tunnel point cloud data. This letter proposes a tunnel surface parameterization algorithm based on a harmonic map, where a 3-D tunnel point cloud is used as a data source to reconstruct a triangle mesh model of the tunnel and then generate a harmonic map depth map of the tunnel inner wall on the triangle mesh. We can obtain the spatial distribution and position information of the appendages and detect whether there are cracks, water leakage, falling pieces, and other diseases by the depth images. The results of this study indicate that the proposed algorithm is suitable for tunnels of various shapes and has low area distortion, which can better avoid the loss of information during dimensionality reduction. Compared with other existing methods, the algorithm has higher efficiency and applicability. [ABSTRACT FROM AUTHOR]

We estimate the dimensions of the spaces of holomorphic sections of certain line bundles to give improved lower bounds on the index of complex isotropic harmonic maps to complex projective space from the sphere and torus, and in some cases from higher genus surfaces. [ABSTRACT FROM AUTHOR]

In this paper we study semi-slant Riemannian maps from Cosymplectic manifolds into Riemannian manifolds. Several fundamental results on integrability of distributions and geometry of foliations are proved for such maps. Also we find the conditions for Riemannian maps to be totally geodesic and investigate some decomposition theorems. Finally, we give some examples of semi-slant Riemannian maps such that the characteristic vector field ζ is either vertical or horizontal. [ABSTRACT FROM AUTHOR]