Theory Of Harmonic Functions Research Articles

Lipschitz regularity of sub-elliptic harmonic maps into CAT(0) space

Abstract We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the 𝑛-dimensional Heisenberg group into CAT ⁡ ( 0 ) \operatorname{CAT}(0) spaces. Our main theorem establishes that these maps have the desired Lipschitz regularity, extending the Hölder regularity in this setting proven in [Y. Gui, J. Jost and X. Li-Jost, Subelliptic harmonic maps with values in metric spaces of nonpositive curvature, Commun. Math. Res. 38 (2022), 4, 516–534] and obtaining same regularity as in [H.-C. Zhang and X.-P. Zhu, Lipschitz continuity of harmonic maps between Alexandrov spaces, Invent. Math. 211 (2018), 3, 863–934] for certain sub-Riemannian geometries; see also [N. Gigli, On the regularity of harmonic maps from RCD ⁢ ( K , N ) \mathrm{RCD}(K,N) to CAT ⁢ ( 0 ) \mathrm{CAT}(0) spaces and related results, preprint (2022), https://arxiv.org/abs/2204.04317; and A. Mondino and D. Semola, Lipschitz continuity and Bochner–Eells–Sampson inequality for harmonic maps from RCD ⁡ ( k , n ) \operatorname{RCD}(k,n) spaces to CAT ⁡ ( 0 ) \operatorname{CAT}(0) spaces, preprint (2022), https://arxiv.org/abs/2202.01590] for the generalisation to RCD spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.

Stability conditions and Teichmüller space

AbstractWe consider a 3-Calabi–Yau triangulated category associated to an ideal triangulation of a marked bordered surface. Using the theory of harmonic maps between Riemann surfaces, we construct a natural map from a component of the space of Bridgeland stability conditions on this category to the enhanced Teichmüller space of the surface. We describe a relationship between the central charges of objects in the category and shear coordinates on the Teichmüller space.

The Harmonic Map Compactification of Teichmüller Spaces for Punctured Riemann Surfaces

In the paper [The Teichmüller theory of harmonic maps, J. Differential Geom. 29 (1989), no. 2, 449–479], Wolf provided a global coordinate system of the Teichmüller space of a closed oriented surface S S with the vector space of holomorphic quadratic differentials on a Riemann surface X X homeomorphic to S S . This coordinate system is via harmonic maps from the Riemann surface X X to hyperbolic surfaces. Moreover, he gave a compactification of the Teichmüller space by adding a point at infinity to each endpoint of harmonic map rays starting from X X in the space. Wolf also showed this compactification coincides with the Thurston compactification. In this paper, we extend the harmonic map ray compactification to the case of punctured Riemann surfaces and show that it still coincides with the Thurston compactification.

Uniqueness theorems for weighted harmonic functions in the upper half-plane

We consider a class of weighted harmonic functions in the open upper half-plane known as α-harmonic functions. Of particular interest is the uniqueness problem for such functions subject to a vanishing Dirichlet boundary value on the real line and an appropriate vanishing condition at infinity. We find that the non-classical case (α ≠ 0) allows for a considerably more relaxed vanishing condition at infinity compared to the classical case (α = 0) of usual harmonic functions in the upper half-plane. The reason behind this dichotomy is different geometry of zero sets of certain polynomials naturally derived from the classical binomial series. These findings shed new light on the theory of harmonic functions, for which we provide sharp uniqueness results under vanishing conditions at infinity along geodesics or along rays emanating from the origin.

Mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with non‐smooth coefficients

AbstractWe prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with low regularity of the coefficients. We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality. We emphasize that our results only rely on the classical theory, and our arguments follow the lines used in the original theory of harmonic functions. We provide two proofs relying on two different formulations of the divergence theorem, one stated for sets with almost C1‐boundary, the other stated for sets with finite perimeter.

Half-harmonic gradient flow: aspects of a non-local geometric PDE

<abstract><p>The goal of this paper is to discuss some of the results in the author's previous papers and expand upon the work there by proving two new results: a global weak existence result as well as a first bubbling analysis for the half-harmonic gradient flow in finite time. In addition, an alternative local existence proof to the one provided in <sup>[<xref ref-type="bibr" rid="b47">47</xref>]</sup> is presented based on a fixed-point argument. This preliminary bubbling analysis leads to two potential outcomes for the possibility of finite-time bubbling until a conjecture by Sire, Wei and Zheng, see <sup>[<xref ref-type="bibr" rid="b40">40</xref>]</sup>, is settled: Either there always exists a global smooth solution to the half-harmonic gradient flow without concentration of energy in finite-time, which still allows for the formation of half-harmonic bubbles as $ t \to +\infty $, or finite-time bubbling may occur in a similar way as for the harmonic gradient flow due to energy concentration in finitely many points. In the first part of the introduction to this paper, we provide a survey of the theory of harmonic and fractional harmonic maps and the associated gradient flows. For clarity's sake, we restrict our attention to the case of spherical target manifolds $ S^{n-1} $, but our discussion extends to the general case after taking care of technicalities associated with arbitrary closed target manifolds $ N $ (cf. <sup>[<xref ref-type="bibr" rid="b48">48</xref>]</sup>).</p></abstract>

$$L^p$$-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions

We establish an optimal $$L^p$$ -regularity theory for solutions to fourth order elliptic systems with antisymmetric potentials in all supercritical dimensions $$n\ge 5$$ : $$\begin{aligned} \Delta ^2 u=\Delta (D\cdot \nabla u)+\text {div}(E\cdot \nabla u) +(\Delta \Omega +G)\cdot \nabla u +f \qquad \ \mathrm{{in}}\ B^n, \end{aligned}$$ where $$\Omega \in W^{1,2}(B^n, so_m)$$ is antisymmetric and $$f\in L^p(B^n)$$ , and $$D, E, \Omega , G$$ satisfy the growth condition (GC-4), under the smallness condition of a critical scale invariant norm of $$\nabla u$$ and $$\nabla ^2 u$$ . This system was brought into lights from the study of regularity of (stationary) biharmonic maps between manifolds by Lamm–Rivière, Struwe, and Wang. In particular, our results improve Struwe’s Hölder regularity theorem to any Hölder exponent $$\alpha \in (0,1)$$ when $$f\equiv 0$$ , and have applications to both approximate biharmonic maps and heat flow of biharmonic maps. As a by-product of our techniques, we also partially extend the $$L^p$$ -regularity theory of approximate harmonic maps by Moser to Rivière-Struwe’s second order elliptic systems with antisymmetric potentials under the growth condition (GC-2) in all dimensions $$n\ge 2$$ when $$p>\frac{n}{2}$$ , which partially confirms an interesting expectation by Sharp.

A Clustering-Based Approach to Automatic Harmonic Analysis: An Exploratory Study of Harmony and Form in Mozart’s Piano Sonatas

We implement a novel approach to automatic harmonic analysis using a clustering method on pitch-class vectors (chroma vectors). The advantage of this method is its lack of top-down assumptions, allowing us to objectively validate the basic music theory premise of a chord lexicon consisting of triads and seventh chords, which is presumed by most research in automatic harmonic analysis. We use the discrete Fourier transform and hierarchical clustering to analyse features of the clustering solutions and illustrate associations between the features and the distribution of clusters over sections of the sonata forms. We also analyse the transition matrix, recovering elements of harmonic function theory.

On O’Hara knot energies I: Regularity for critical knots

We develop a regularity theory for extremal knots of scale invariant knot energies defined by J. O’Hara in 1991. This class contains as a special case the Möbius energy. For the Möbius energy, due to the celebrated work of Freedman, He, and Wang, we have a relatively good understanding. Their approach is crucially based on the invariance of the Möbius energy under Möbius transforms, which fails for all the other O’Hara energies. We overcome this difficulty by re-interpreting the scale invariant O’Hara knot energies as a nonlinear, nonlocal $L^p$-energy acting on the unit tangent of the knot parametrization. This allows us to draw a connection to the theory of (fractional) harmonic maps into spheres. Using this connection we are able to adapt the regularity theory for degenerate fractional harmonic maps in the critical dimension to prove regularity for minimizers and critical knots of the scale-invariant O’Hara knot energies.

Extension of Krust theorem and deformations of minimal surfaces

In the minimal surface theory, the Krust theorem asserts that if a minimal surface in the Euclidean 3-space \(\mathbb {E}^3\) is the graph of a function over a convex domain, then each surface of its associated family is also a graph. The same is true for maximal surfaces in the Minkowski 3-space \(\mathbb {L}^3\). In this article, we introduce a new deformation family that continuously connects minimal surfaces in \(\mathbb {E}^3\) and maximal surfaces in \(\mathbb {L}^3\), and prove a Krust-type theorem for this deformation family. This result induces Krust-type theorems for various important deformation families containing the associated family and the López-Ros deformation. Furthermore, minimal surfaces in the isotropic 3-space \(\mathbb {I}^3\) appear in the middle of the above deformation family. We also prove another type of Krust’s theorem for this family, which implies that the graphness of such minimal surfaces in \(\mathbb {I}^3\) strongly affects the graphness of deformed surfaces. The results are proved based on the recent progress of planar harmonic mapping theory.

Computing harmonic maps between Riemannian manifolds

AbstractIn our previous paper (Gaster et al., 2018, arXiv:1810.11932), we showed that the theory of harmonic maps between Riemannian manifolds, especially hyperbolic surfaces, may be discretized by introducing a triangulation of the domain manifold with independent vertex and edge weights. In the present paper, we study convergence of the discrete theory back to the smooth theory when taking finer and finer triangulations, in the general Riemannian setting. We present suitable conditions on the weighted triangulations that ensure convergence of discrete harmonic maps to smooth harmonic maps, introducing the notion of (almost) asymptotically Laplacian weights, and we offer a systematic method to construct such weighted triangulations in the two-dimensional case. Our computer software Harmony successfully implements these methods to compute equivariant harmonic maps in the hyperbolic plane.

Spherical conical metrics and harmonic maps to spheres

A spherical conical metric g g on a surface Σ \Sigma is a metric of constant curvature 1 1 with finitely many isolated conical singularities. The uniformization problem for such metrics remains largely open when at least one of the cone angles exceeds 2 π 2\pi . The eigenfunctions of the Friedrichs Laplacian Δ g \Delta _g with eigenvalue λ = 2 \lambda =2 play a special role in this problem, as they represent local obstructions to deformations of the metric g g in the class of spherical conical metrics. In the present paper we apply the theory of multivalued harmonic maps to spheres to the question of existence of such eigenfunctions. In the first part we establish a new criterion for the existence of 2 2 -eigenfunctions, given in terms of a certain meromorphic data on Σ \Sigma . As an application we give a description of all 2 2 -eigenfunctions for metrics on the sphere with at most three conical singularities. The second part is an algebraic construction of metrics with large number of 2 2 -eigenfunctions via the deformation of multivalued harmonic maps. We provide new explicit examples of metrics with many 2 2 -eigenfunctions via both approaches, and describe the general algorithm to find metrics with arbitrarily large number of 2 2 -eigenfunctions.

Characterizations of Product Hardy Spaces in Bessel Setting

In this paper, we work in the setting of Bessel operators and Bessel Laplace equations studied by Weinstein, Huber, and the harmonic function theory in this setting introduced by Muckenhoupt–Stein, especially the generalised Cauchy–Riemann equations and the conjugate harmonic functions. We provide the equivalent characterizations of product Hardy spaces associated with Bessel operators in terms of the Bessel Riesz transforms, non-tangential and radial maximal functions defined via Poisson and heat semigroups, based on the atomic decomposition, the generalised Cauchy–Riemann equations, the extension of Merryfield’s result which connects the product non-tangential maximal function and area function, and on the grand maximal function technique which connects the product non-tangential and radial maximal function. We then obtain directly the decomposition of the product BMO space associated with Bessel operators.

Pseudo-holomorphic curves: A very quick overview

AbstractThis is a review article on pseudo-holomorphic curves which attempts at touching all the main analytical results. The goal is to make a user friendly introduction which is accessible to those without an analytical background. Indeed, the major accomplishment of this review is probably its short length.Nothing in here is original and can be found in more detailed accounts such as [6] and [8]. The exposition of the compactness theorem is somewhat different from that in the standard references and parts of it are imported from harmonic map theory [7], [5]. The references used are listed, but of course any mistake is my own fault.

Optimal estimates for hyperbolic Poisson integrals of functions in [formula omitted] with [formula omitted] and radial eigenfunctions of the hyperbolic Laplacian

Assume that p∈(1,∞] and u=Ph[ϕ], where ϕ∈Lp(Sn−1,Rn). Then for any x∈Bn, we obtain the sharp inequalities |u(x)|≤Cq1q(x)(1−|x|2)n−1p‖ϕ‖Lpand|u(x)|≤Cq1q(1−|x|2)n−1p‖ϕ‖Lp for some function Cq(x) and constant Cq in terms of Gauss hypergeometric and Gamma functions, where q is the conjugate of p. This result generalizes and extends some known results from harmonic mapping theory (Kalaj and Marković (2012, Theorems 1.1 and 1.2) and Axler et al. (1992, Proposition 6.16)). The proofs are mainly based on certain characterizations of the radial eigenfunctions of the hyperbolic Laplacian Δh, which are of independent interest.

Harmonic surfaces in the Cayley plane

We consider the twistor theory of nilconformal harmonic maps from a Riemann surface into the Cayley plane $\mathbf{O} P^2=F_4/\mathrm{Spin}(9)$. By exhibiting this symmetric space as a submanifold of the Grassmannian of $10$-dimensional subspaces of the fundamental representation of $F_4$, techniques and constructions similar to those used in earlier works on twistor constructions of nilconformal harmonic maps into classical Grassmannians can also be applied in this case. The originality of our approach lies on the use of the classification of nilpotent orbits in Lie algebras as described by D. Djokovic.

Harmonic maps between ideal 2-dimensional simplicial complexes

We prove existence and regularity results for energy minimizing maps between ideal hyperbolic 2-dimensional simplicial complexes. The spaces in question were introduced by Charitos–Papadopoulos, who describe their Teichmuller spaces and some compactifications. This work is a first step in introducing harmonic map theory into the Teichmuller theory of these spaces.

Method of Riemann surfaces for an inverse antiplane problem in an n-connected domain

ABSTRACTAn inverse problem of the theory of harmonic functions for an n-connected domain is analyzed. The problem is equivalent to a problem of antiplane elasticity on determination of the profiles of n uniformly stressed inclusions. The inclusions are in ideal contact with the surrounding matrix, the stress field inside the inclusions is uniform, and at infinity the body is subjected to antiplane uniform shear. The exterior of the inclusions, an n-connected domain, is treated as the image by a conformal map of an n-connected slit domain with the slits lying in the same line. The inverse problem is solved by quadratures by reducing it to two Riemann-Hilbert problems on a Riemann surface of genus n−1. Samples of two and three symmetric and non-symmetric uniformly stressed inclusions are reported.

Stress energy tensor of C-stationary maps

We consider a functional Econf(f) on the space of smooth maps f between Riemannian manifolds, such that f is a weakly conformal map if and only if Econf(f) = 0. The quantity Econf(f) is a measure of weak conformality of maps f. Stationary maps for this functional are called C-stationary maps (Nakauchi, 2011, Kawai and Nakauchi, 2013). Weakly conformal maps are always C-stationary ones. On the other hand, in the field of calculus of variations on Riemannian manifolds, the theory of harmonic maps is developed with applications to other fields. (See two reports by Eells and Lemaire, 1978, Eells and Lemaire, 1988.) Several concepts are given in this theory. The stress energy tensor of harmonic maps is first defined by Baird and Eells (1981). This tensor is a conserved quantity. In this paper we introduce a stress energy tensor of C-stationary maps and give some results.

Higgs bundles over cell complexes and representations of finitely presented groups

The purpose of this paper is to extend the Donaldson-Corlette theorem to the case of vector bundles over cell complexes. We define the notion of a vector bundle and a Higgs bundle over a complex, and describe the associated Betti, de Rham and Higgs moduli spaces. The main theorem is that the $SL(r, \mathbb{C})$ character variety of a finitely presented group $\Gamma$ is homeomorphic to the moduli space of rank $r$ Higgs bundles over an admissible complex $X$ with $\pi_1(X) = \Gamma$. A key role is played by the theory of harmonic maps defined on singular domains.