Time-Harmonic Solutions for Maxwell’s Equations in Anisotropic Media and Bochner–Riesz Estimates with Negative Index for Non-elliptic Surfaces

We solve time-harmonic Maxwell’s equations in anisotropic, spatially homogeneous media in intersections of $$L^p$$ L p -spaces. The material laws are time-independent. The analysis requires Fourier restriction–extension estimates for perturbations of Fresnel’s wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds but without flat points, and surfaces with conical singularities. Our estimates are based on new Bochner–Riesz estimates with negative index for non-elliptic surfaces.

Isoperimetric inequalities and geometry of level curves of harmonic functions on smooth and singular surfaces

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 $$L'$$ L ′ and $$L''$$ 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.

Uniformization of branched surfaces and Higgs bundles

Given a compact connected Riemann surface [Formula: see text] of genus [Formula: see text], and an effective divisor [Formula: see text] on [Formula: see text] with [Formula: see text], there is a unique cone metric on [Formula: see text] of constant negative curvature [Formula: see text] such that the cone angle at each point [Formula: see text] is [Formula: see text] [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 [Formula: see text] corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on [Formula: see text] parametrized by a nonempty open subset of [Formula: see text] 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 [Formula: see text].

Supersymmetric black holes with spiky horizons

We use the recipe of [1] to find half-BPS near-horizon geometries in the t3 model of N = 2, D = 4 gauged supergravity, and explicitely construct some new examples. Among these are black holes with noncompact horizons, but also with spherical horizons that have conical singularities (spikes) at one of the two poles. A particular family of them is extended to the full black hole geometry. Applying a double-Wick rotation to the near-horizon region, we obtain solutions with NUT charge that asymptote to curved domain walls with AdS3 world volume. These new solutions may provide interesting testgrounds to address fundamental questions related to quantum gravity and holography.

Smooth bubbling geometries without supersymmetry

We construct the first smooth bubbling geometries using the Weyl formalism. The solutions are obtained from Einstein theory coupled to a two-form gauge field in six dimensions with two compact directions. We classify the charged Weyl solutions in this framework. Smooth solutions consist of a chain of Kaluza-Klein bubbles that can be neutral or wrapped by electromagnetic fluxes, and are free of curvature and conical singularities. We discuss how such topological structures are prevented from gravitational collapse without struts. When embedded in type IIB, the class of solutions describes D1-D5-KKm solutions in the non-BPS regime, and the smooth bubbling solutions have the same conserved charges as a static four-dimensional non-extremal Cvetic-Youm black hole.

Rotating multi-charge spindles and their microstates

Some AdS3 × M7 type IIB vacua have been recently proposed to arise from D3-branes wrapped on a spindle, a sphere with conical singularities at the poles. We explicitly construct a generalization of these solutions corresponding to a class of electrically charged and rotating supersymmetric black strings in AdS5 × S5 with general magnetic fluxes on the spindle. We then perform a counting of their microstates using the charged Cardy formula. To this purpose, we derive the general form of the anomaly polynomial of the dual $$ \mathcal{N} $$ N = (0, 2) CFT in two dimensions and we show that it can be obtained via a simple gluing procedure.

Real classical geometry with arbitrary deficit parameter(s) $$\alpha (_{I})$$ in deformed Jackiw–Teitelboim gravity

An interesting deformation of Jackiw–Teitelboim (JT) gravity has been proposed by Witten by adding a potential term $$U(\phi )$$ U ( ϕ ) as a self-coupling of the scalar dilaton field. During calculating the path integral over fields, a constraint comes from integration over $$\phi $$ ϕ as $$R(x)+2=2\alpha \delta (\vec {x}-\vec {x}')$$ R ( x ) + 2 = 2 α δ ( x → - x → ′ ) . The resulting Euclidean metric suffered from a conical singularity at $$\vec {x}=\vec {x}'$$ x → = x → ′ . A possible geometry is modeled locally in polar coordinates $$(r,\varphi )$$ ( r , φ ) by $$\mathrm{d}s^2=\mathrm{d}r^2+r^2\mathrm{d}\varphi ^2,\varphi \cong \varphi +2\pi -\alpha $$ d s 2 = d r 2 + r 2 d φ 2 , φ ≅ φ + 2 π - α . In this letter we show that there exists another family of ”exact” geometries for arbitrary values of the $$\alpha $$ α . A pair of exact solutions are found for the case of $$\alpha =0$$ α = 0 . One represents the static patch of the AdS and the other one is the non-static patch of the AdS metric. These solutions were used to construct the Green function for the inhomogeneous model with $$\alpha \ne 0$$ α ≠ 0 . We address a type of phase transition between different patches of the AdS in theory because of the discontinuity in the first derivative of the metric at $$x=x'$$ x = x ′ . We extended the study to the exact space of metrics satisfying the constraint $$R(x)+2=2\sum _{i=1}^{k}\alpha _i\delta ^{(2)}(x-x'_i)$$ R ( x ) + 2 = 2 ∑ i = 1 k α i δ ( 2 ) ( x - x i ′ ) as a modulus diffeomorphisms for an arbitrary set of deficit parameters $$(\alpha _1,\alpha _2,\ldots ,\alpha _k)$$ ( α 1 , α 2 , … , α k ) . The space is the moduli space of Riemann surfaces of genus g with k conical singularities located at $$x'_k$$ x k ′ , denoted by $$\mathcal {M}_{g,k}$$ M g , k .

AdS3 from M-branes at conical singularities

M-theory is known to possess supersymmetric solutions where the geometry is AdS3 × S3 × S3 warped over a Riemann surface Σ2. The simplest examples in this class can be engineered by placing M2 and M5 branes as defects inside of a stack of background M5 branes. In this paper we show that a generalization of this construction yields more general solutions in the aforementioned class. The background branes are now M5’s carrying M2 brane charge, while the defect branes are now placed at the origin of a flat hyperplane with a conical defect. The equations of motion imply a relation between the deficit angle produced by the conical defect and the M2 charge carried by the background branes.