Papers

For noncompact smooth Kähler-Ricci shrinkers, we prove a Matsushima theorem identifying the centralizer of the soliton field with the complexification of its Killing centralizer, uniqueness up to biholomorphism on a fixed complex manifold, and K-polystability of the associated polarized Fano fibration.

We introduce a global frequency function for harmonic functions on rotationally symmetric Cartan-Hadamard manifolds. In real hyperbolic space, we also identify a different frequency function associated with the normalized spherical $L^2$-mass of a hyperbolic harmonic function and prove its monotonicity. As a consequence, we obtain a logarithmic convexity theorem for the normalized spherical $L^2$-mass and a hyperbolic analog of the classical three-sphere theorem.

In this article, we give a global construction of the Virasoro conformal blocks on Teichmüller space of Riemann surfaces of genus $g$ with $m$ marked points, when the central charge is $c>25$. We prove that they are global holomorphic sections of a holomorphic line bundle and they satisfy the Ward identity, which encodes their conformal invariance. For the sphere with $3$ points, the space of conformal blocks is one-dimensional. For a given marked Riemann surface and a marked pair of pants decomposition, it is in general an infinite dimensional Hilbert space, isomorphic to $L^2(\R_+^{3g-3+m})$ with respect to some measure involving the DOZZ structure constants. We show that it carries a projective unitary representation of the mapping class group. More generally, there are unitary isomorphisms for each Moore-Seiberg move, changing a marked pair of pants decomposition into another one.

This paper is the second part of the proof of the conformal bootstrap for Liouville conformal field theory on compact surfaces with boundary. It is devoted to the spectral theory of the half-annulus semigroup and to the resulting bootstrap formula. Building on the Segal axioms and gluing properties established in Part~1~\cite{GRW1}, we identify the generator of the half-annulus semigroup with the boundary Hamiltonian and establish its spectral decomposition by scattering methods. This yields a direct-integral decomposition of the boundary state space into irreducible Virasoro representations, governed by the boundary spectral measure. Combining this decomposition with Ward identities for boundary deformations, we express general correlation functions as integrals over spectral parameters attached to cutting curves, with integrands given by products of bulk and boundary structure constants and the corresponding conformal blocks. These results also underpin the analyticity and symmetry properties of conformal blocks~\cite{GRSSbloc}, and contribute to the construction of mapping class group representations on spaces of conformal blocks~\cite{PapierBlocs}.

The generalization of the Kähler metric in the complex Finsler setting admits four natural versions. Among these, the strongly Kähler–Finsler condition is known to coincide with one of the intermediate conditions. In this paper, we establish the equivalence between the weakly Kähler–Finsler condition and the remaining intermediate condition. Moreover, we obtain new necessary and sufficient conditions for a weakly Kähler–Finsler metric to be strongly Kähler–Finsler. Using these criteria, we construct, for the first time, smooth examples of weakly Kähler–Finsler metrics which are not Kähler–Finsler, via the Kobayashi metrics on smooth bounded strongly convex domains.

Given a Finsler manifold $(M,F)$ and suitable nested regions $B_1\subset B_2\subset B_3$, with $M\setminus B_1$ connected and non-contractible, as in the model case of concentric balls in $\mathbb R^n$, $n\geq2$, we consider conformal deformations $\varphi_\lambda F$ which can be chosen so as to leave $F$ unchanged outside $B_3$ and grow on $B_2\setminus B_1$. This growth creates an energy barrier, diverging with $\lambda$, that low-energy curves cannot cross. The minimax families can therefore be chosen in the path space of $M\setminus B_1$, which contains compact subsets of arbitrarily large category. Consequently, for every $m\in\mathbb N$ and all sufficiently large $\lambda$, there exist at least $m$ geodesics joining two prescribed points, all avoiding $B_1$. This extends to the Finsler setting an idea of Capozzi, Fortunato and Greco. The same scheme applies to Randers metrics $\alpha+\beta$, with $\alpha$ deformed and $\beta$ unchanged, as well as to fixed-energy trajectories of non-relativistic electromagnetic systems and relativistic Lorentz-force trajectories, and light rays in spacetimes endowed with a causal Killing field. In the non-relativistic case only the scalar potential is perturbed, while the magnetic field is preserved. When the Killing field is not everywhere timelike, the Fermat principle yields a possibly singular Randers–Kropina metric on a spacelike hypersurface. Under a natural admissible category condition, combining localized barriers with Randers approximation gives $m$ Randers–Kropina geodesics with pairwise distinct lengths, equivalently $m$ future-pointing light rays with pairwise distinct arrival times.

Let $M^n$, $n\ge3$, be the smooth, connected, complete, noncompact boundary of a closed convex set with nonempty interior in $\mathbb{R}^{n+1}$. We prove that its normalized scalar-curvature integral at infinity is at most $4\pi\omega_{n-2}(1-\operatorname{AVR}(M))$, except when, up to an ambient rigid motion, $M$ is the product of a smooth closed convex surface in $\mathbb{R}^3$ and $\mathbb{R}^{n-2}$. These exceptional products have zero asymptotic volume ratio and limit $8\pi\omega_{n-2}$. We also characterize the equality case for the $4\pi\omega_{n-2}(1-\operatorname{AVR}(M))$ bound. The proof uses exterior tube volumes together with the cone Steiner and Gauss–Bonnet formulas. An elementary comparison of intrinsic and Euclidean radial distances proves existence of the intrinsic limit and identifies its exact value from the cone coefficients. The same conclusions extend to orientable convex submanifolds of arbitrary codimension

We give a detailed and complete algebraic proof of Simons' Riemannian holonomy theorem by a single strong induction on the dimension of the linear span of the curvature orbit. The proof uses the construction of flats and root centralizers, but replaces the detailed analysis of the common zero-weight space, the most technical core of previous algebraic proofs, by a trace-vanishing lemma. The lemma applies to curvature operators contained in an ideal that annihilates the curvature module. Restriction to a common geodesic subspace produces a proper invariant kernel. Induction makes this kernel fixed by the holonomy action, after which the trace lemma shows that the root centralizers span the ambient space. Their intersection recovers the maximal flat, and irreducibility completes the proof. For the completeness we also include the detailed derivation of the local symmetry of the underlying Riemannian manifold from the algebraic theorem.

To an oriented quiver with boundary $\Gamma$ and a compact connected Lie group $G$ we associate a Nahm-Kirchhoff moduli space $\mathcal{M}_{\mathbb{H}}(\Gamma)$: solutions of Nahm's equations on the edges with Kirchhoff matching at interior vertices, modulo boundary-trivial gauge transformations. For connected $\Gamma$, $\mathcal{M}_{\mathbb{H}}(\Gamma)$ is a smooth manifold of dimension $4(|E|-|\Gamma_{\mathrm{int}}|)\dim G$ with a hyperkähler structure arising as an infinite-dimensional hyperkähler quotient. Via a Kempf-Ness argument on the Cartan-Hadamard manifold $G_{\mathbb{C}}/G$, we prove a quiver analogue of Donaldson's theorem identifying it with the complex-symplectic quotient $\mathcal{M}_{\mathbb{C}}(\Gamma)$ of the complex Nahm equations by the complexified gauge group. This quotient is independent of edge lengths and isomorphic to a finite-dimensional complex-symplectic quotient of $(T^*G_{\mathbb{C}})^E$, while the hyperkähler metric depends on them: uniform rescaling acts by a homothety, and the transported metrics genuinely vary. By cutting and gluing, $\Gamma\mapsto\mathcal{M}_{\mathbb{C}}(\Gamma)$ is a 2D TQFT valued in complex Hamiltonian manifolds, à la Moore-Tachikawa, whereas the hyperkähler refinement is not functorial on $\mathbf{Cob}_2$: spines of a cylinder with different lengths give non-isomorphic spaces. Retaining lengths, these spaces form a metric field theory, a functor of non-unital monoidal categories from metric cobordisms to hyperkähler manifolds composed by hyperkähler reduction; over each cone of $M_{g,n}^{\mathrm{trop}}$ they give a family with fixed complex-symplectic fibre and varying metric. Finally, harmonic gauge representatives converge as an internal edge collapses, and on a long edge with regular semisimple Higgs field the Kempf-Ness minimizer is exponentially confined near its centralizer; the limiting metrics are left open.

We prove that metric measure spaces arising from $2$-manifolds equipped with Riemannian metrics $g$ such that $g, g^{-1} \in L^\infty_{\rm loc} \cap W^{1,p}_{\rm loc}$ are infinitesimally Hilbertian. Along the way, we establish uniform lower $1$-capacity bounds of several $1$-dimensional subsets of $\R^2$.

For a closed orientable surface of genus at least 2, we study the large-scale geometry of the (symmetrized) Lipschitz metric on its space of projective filling geodesic currents. We show that the rank and, hence, asymptotic dimension of the space of filling geodesic currents is infinite. This gives the first example of a natural proper space associated to a closed surface, with a metrically proper isometric action of the mapping class group, that has infinite asymptotic dimension. In contrast, we show that any finite simplex of geodesic currents has rank uniformly bounded above by $8g-9$. We then show a sufficient criterion for its rank and asymptotic dimension to be equal to $8g-9$ and construct examples satisfying it, thus showing the bound is sharp. We further prove that Teichmüller space equipped with the Lipschitz metric has rank equal to $3g-3$. In particular, the space of projective filling geodesic currents and the Teichmüller space are not quasi-isometric.

We construct Riemannian metrics of strictly positive sectional curvature on at least 508 pairwise nondiffeomorphic exotic fifteen-spheres, without distinguishing orientations. The construction starts with quaternionic line bundles over the product of the quaternionic projective plane and the four-sphere. Surgery on an eleven-sphere fiber produces homotopy fifteen-spheres with conjugation-equivariant attaching maps. A calculation on the surgery filling gives their Eells–Kuiper invariants and distinguishes 1015 nonstandard oriented smooth types. The required representation has infinitesimal action norm at most one, so the examples also satisfy the hypothesis of Galaz-García's bounded-representation transfer theorem. The topological construction and invariant calculation are independent of the curvature input.

We derive the sub-Riemannian geodesics of the affine-additive group using the Pontryagin Maximum Principle (PMP) and compute the associated cut times. We then proceed to construct the Carnot-Carathéodory sphere of the affine-additive group as well as the flask surface; the latter is the analogue of the Pansu sphere (bubble set) in the first Heisenberg group.

We construct a finite cyclic $L_3$ presentation (that is, a cyclic $L_\infty$ algebra with brackets vanishing above arity three) of the variational deformation theory defined by the heterotic $G_2$ superpotential near a torsion-free standard embedding, to first order in $\alpha'$. Auxiliary fields for the inverse coframe, spinor derivative, flux and induced tangent connection give a shifted-cotangent Hamiltonian of degree at most four. Eliminating the auxiliary fields recovers the physical functional. On the flat seven-torus, this functional has a nonzero quintic Taylor coefficient in local physical spinor coordinates. For $Y=S^1\times X$, with $X$ a Calabi–Yau threefold, we restrict to the product locus with horizontal gauge and gerbe data. The functional reduces to $ \frac L4\operatorname{Im}\int_X(H+i\,d\omega)\wedge\Omega, $ where $L$ is the circle length and $\Omega=e^{-2\Phi}\Psi$ is the weighted complex volume. The induced $SU(3)$ structure selects a Hermitian tangent Courant subalgebroid. Transverse isotropic lifts in its complexification encode the holomorphic connection and string equations, together with the induced tangent connection, weighted canonical line and gerbe hierarchy. Their Taylor brackets form an $L_3$ algebra whose string part agrees with the known holomorphic heterotic algebra. We then show that this holomorphic theory is obtained by characteristic Hamiltonian reduction of a finite contractible stabilization of the product variational theory, in every cochain degree. While the quotient captures the holomorphic equations, conformal balance, gauge primitivity and leading-flux admissibility are additional conditions on a real representative.

We determine which homotopy nine-spheres admit positive sectional curvature: they are precisely the four smooth types with vanishing real Dirac index. The other four admit no metric of nonnegative sectional curvature. In particular, we construct a strictly positively curved metric on the exotic Kervaire nine-sphere, answering the longstanding nonnegative-curvature existence question in this dimension. The proof keeps explicit attaching maps throughout. A quaternionic commutator realizes the Kervaire sphere, a mixed representation realizes the Hopf image of the exotic eight-sphere, and a four-periodic normal-framing formula identifies the remaining index-zero class. Following the compatible-disk method, we give the curvature and boundary estimates for the first two models, including an explicit lower bound for the Kervaire metric in terms of the curvature and first covariant-derivative norms of a fixed connection, with the base scale and fiber normalization fixed. For the third model, we construct a positive metric on a cylinder joining a quotient metric to its pullback by a marked right Hopf twist. A four-form modification controls the connection terms involving both left and right quaternionic multiplication. Two positive disk fillings close the cylinder with the prescribed attaching map. A local convolution lemma supplies the smooth gluing while preserving a strict sectional-curvature lower bound.

It is well known that infinitesimal isometries preserve many of the canonical geometric structures of a Riemannian or Finslerian manifold. In this paper, we identify those geometric structures of a Finslerian manifold that are preserved by an infinitesimal isometry (a Killing vector field) and whose preservation characterises such fields. Using the Frölicher-Nijenhuis formalism as a unified geometric framework, we establish twelve equivalent characterisations of the Killing property in the Finslerian setting, utilizing the Finsler norm, the sphere bundle, contact and symplectic structures, Reeb and Hamiltonian vector fields, first integrals, the homogenized almost complex structure, and the Sasaki metrics. Furthermore, we derive the local expressions of these characterisations, yielding several equivalent forms of the Killing equations, including scalar, one-form and tensorial formulations. In our approach, none of these formulations is regarded as primary; each reflects a different geometric aspect of the Killing condition.

We consider the problem of uniqueness of Ricci flows, starting from singular initial data. Even when the initial data is smooth and smoothly obtained, in the complete noncompact case uniqueness is not known in full generality. When starting from nonsmooth initial data, very little is known, even in the compact case. We obtain a uniqueness result for Ricci flows with scaling-invariant curvature bounds, a suitable noncollapsing assumption, and sectional curvature lower bounds, coming out of (possibly singular) Reifenberg spaces. As a consequence of our result, and existence results obtained by Simon–Topping and Lai, there is a canonical way to define a smooth structure on any Reifenberg uniformly volume-noncollapsed PIC2 limit space. Further, we obtain a diffeomorphic version of Perelman's stability theorem for these limit spaces.

We show that the conformal class of a compact spin manifold with boundary is uniquely determined by its boundary values of harmonic spinors if the metric is locally conformal to a real-analytic metric in some coordinates. More precisely, we consider the boundary conjugation (BC) map for the Dirac operator corresponding to MIT boundary conditions, and prove that if two locally conformally real-analytic (LCRA) metrics have equivalent BC maps over an open subset of the boundary, then the two manifolds are conformally equivalent. The class of LCRA manifolds includes conformally Einstein manifolds, as well as all smooth surfaces. We also show that a real-analytic unitary connection on an LCRA manifold is uniquely determined up to local gauge equivalence by the BC map of its twisted Dirac operator in dimensions greater than 2.

We develop a representation-theoretic approach to determine the Morse index and nullity of an embedded minimal surface based only on its topology and symmetry group. The method combines an equivariant comparison between the second variations of area and energy with the representation theory of the dihedral group. In the Euclidean unit ball we compute for every $k\geq3$ the Morse index and nullity of any free boundary minimal $k$-noid with prismatic symmetry of order $4k$, and of any free boundary minimal surface of genus one with connected boundary and antiprismatic symmetry of order eight. In the round sphere we obtain an alternative computation for the Morse index and nullity for the first family of the Lawson surfaces, using only half of their symmetry group. The results are independent of any particular construction of the surfaces.

We construct hypocoercive Kolmogorov-type operators on the Euclidean motion group $\mathrm{SE}(3)=\mathbb{R}^3\rtimes\mathrm{SO}(3)$ using ZXZ Euler angles and study their long-time behaviour using the abstract hypocoercivity method. The operator $L=S-A$ decomposes into a symmetric part $S$, corresponding to a degenerate diffusion on the rotation component $\mathrm{SO}(3)$, and an antisymmetric part $A$, which captures transport and torque effects induced by the Lie group structure. The associated invariant probability measure is $\mu_\Phi=\gamma_\Phi\otimes\nu_3$, where $\gamma_\Phi$ is a probability measure on $\mathbb{R}^3$ with density $Z(\Phi)^{-1}e^{-\Phi}$ with respect to the Lebesgue measure, and $\nu_3$ is the normalized Haar measure on $\mathrm{SO}(3)$. As a consequence, if $\Phi\in C^\infty(\mathbb{R}^3)$ is bounded from below, $\gamma_\Phi$ satisfies a Poincaré inequality with constant $\Lambda>0$, and there exists $c<\infty$ such that \begin{equation*} |\nabla_\xi^2\Phi(\xi)| \leq c\bigl(1+|\nabla_\xi\Phi(\xi)|\bigr), \quad \xi\in\mathbb{R}^3, \end{equation*} then the semigroup generated by the closure of $L$ converges exponentially fast to equilibrium in $L^2(\mathrm{SE}(3),\mu_\Phi)$.

Branson's $Q$-curvature is a natural fourth-order counterpart of scalar curvature in conformal geometry. In analogy with the classical recovery of the Ricci tensor from the linearized scalar curvature, Lin and Yuan introduced a symmetric $2$-tensor $J_g$ through the formal adjoint of the linearized $Q$-curvature and established its explicit decomposition in terms of the Bach tensor and auxiliary curvature terms. In this paper, we study the associated fourth-order geometric flow $\partial_t g = -2J_g$ on closed Riemannian manifolds of dimension $n \ge 4$. We establish short-time existence and uniqueness using a biharmonic DeTurck gauge and derive global and local integral smoothing estimates under curvature bounds. In dimension four, we prove long-time existence assuming a uniform Sobolev inequality and sufficiently small total $L^2$ curvature energy throughout the maximal interval of existence. Under uniform curvature and Sobolev bounds, time integrability of the $L^2$-norm of $J$ implies smooth convergence in fixed coordinates to a $J$-flat metric. Explicit Einstein and product solutions illustrate the evolution and show that small curvature energy alone does not prevent degeneration. Positive initial Yamabe constant and positive total $Q$-curvature give a uniform positive Yamabe lower bound throughout the smooth four-dimensional flow, whereas in higher dimensions total $Q$-curvature can decrease even near flat metrics. Finally, we prove that every closed four-dimensional $J$-soliton is trivial, its metric is Bach-flat with constant $Q$-curvature, and its soliton vector field is Killing.

We characterize lower curvature bounds on metric surfaces by the Bishop–Gromov inequality for metric balls.

We study the decomposed Möbius energies $E_1$ and $E_2$ at the round circle and determine their complete normal Hessians: $H_1(C)=2\pi(|D_s|^3-|D_s|)I_2$ and $H_2(C)=-(4\pi/3)(|D_s|^3-|D_s|)I_2$. Thus $H_1(C)=3H_E(C)$ and $H_2(C)=-2H_E(C)$, where $H_E(C)$ is the Hessian of the full Möbius energy. This exact proportionality singles out the Möbius-invariant combination $F=2E_1+3E_2+2\pi^2$. We prove that $F(C)=0$ and that its first three derivatives vanish on the full space of parametrized variations. For scalar binormal variations, the fourth variation has a Fourier tensor with complete $3+1$ resonance cancellation and finite $2+2$ fixed-sum blocks. Their positivity follows from an explicit weighted-difference tridiagonalization and a closed determinant formula. For arbitrary normal variations in $\mathbb{R}^3$, Möbius invariance in $\mathbb{R}^4$ reduces the radial component to a second binormal direction. The resulting two-component quartic form splits into trace, symmetric-traceless, and antisymmetric blocks, which are positive on the active coordinates. Consequently, $D^4F(C)[u,u,u,u]\geq 0$ for every smooth real normal field $u$, with equality exactly on the normal tangent space of the Möbius family of round circles. This yields a fourth-order nondegeneracy phenomenon that is not determined by the total-energy Hessian alone.

The space of $n\times n$ density matrices is the full state space of an $n$-level quantum system. While it has been asserted in the literature on quantum information theory that decomposing this state space into equirank pieces yields a Whitney stratification, rigorous proofs have remained absent. Using methods from invariant theory, this work provides a proof that the rank decomposition of the matrix algebra constitutes a (b)-regular stratification in the sense of Whitney. Furthermore, we demonstrate how the Uhlmann connection - which describes the adiabatic holonomy of mixed states and reduces to the Berry connection on pure states - can be geometrically treated as a stratified connection on a stratified vector bundle. Finally, our appendix provides a self-contained exposition of stratified fiber bundles and the new concept of stratified connections, which may be of independent interest.

Problem 73 on Green's list of open problems asks whether a smooth submanifold of codimension two in $\mathbb{R}^n$ must meet some affine two-plane in five points when $n$ is large. In this paper, we answer this question for every $n \geq 6$. Every nonempty relatively open subset of such a submanifold contains the vertices of an affinely regular pentagon, and the bound $n \geq 6$ is sharp for affinely regular pentagons. More generally, for every prime $p \geq 5$, the graph of a continuous map from a ball in $\mathbb{R}^{p-1}$ to $\mathbb{R}^2$ contains an affinely regular $p$-gon. A second argument gives six coplanar points when $n \geq 7$. Both proofs find the points by making the low Fourier modes of the graph map vanish along a circle.

Let $G$ be a compact connected semisimple Lie group, let $Q$ be a bi-invariant metric and let $\rho$ be its Cartan three-form. We prove that every left-invariant generalized Ricci flow with $[H_0]=[\rho]$ exists for all positive time and converges smoothly to a Bismut-Ricci-flat pair without rescaling or pullback. The standard pair uniquely maximizes generalized scalar curvature in its torsion class, and every left-invariant Bismut-Ricci-flat pair on a compact simple group is standard, answering questions of Fusi, Lafuente and Stanfield and of Lauret and Will. On the product of two copies of a compact simple group, nonstandard stationary pairs in the Cartan class are known; we construct nonstationary flows with the exact initial form $[H_0]=[\rho]$ and nonstandard limits, disproving the general semisimple case of the convergence conjecture of Garcia-Fernandez and Streets. The proofs use a proper nonincreasing energy and a decomposition of Killing fields that parametrizes the stationary pairs by nilpotent Lie algebra endomorphisms.

We study almost Hermitian manifolds whose lowered Nijenhuis tensor is totally skew-symmetric and whose characteristic (Bismut) torsion is closed, the almost-SKT condition. This extends the SKT (pluriclosed) condition beyond the integrable setting. For a fixed almost complex structure, these constraints are linear in the metric. We introduce an overdetermined ellipticity condition which, on compact manifolds, makes the admissible infinitesimal deformation space finite-dimensional and yields short-time existence for a constrained Bismut–Ricci flow. In real dimension six, almost-SKT structures satisfy a dichotomy: either the almost complex structure is integrable, or the skew Nijenhuis tensor is nowhere vanishing with constant norm and the Bismut Ricci form vanishes. Hence, on compact non-integrable almost-SKT six-manifolds, the constrained Bismut–Ricci flow is stationary. In dimension eight, ellipticity always fails, while in higher dimensions Nijenhuis rigidity holds on an open dense locus. In arbitrary dimension, we characterize when total skew-symmetry of the lowered Nijenhuis tensor determines the compatible metric pointwise up to scale. On compact manifolds, with closed torsion, this implies uniqueness of the almost-SKT metric up to scaling; the constrained Bismut–Ricci flow is then an explicit homothetic solution. We also relate this flow to generalized Ricci flow and characterize the infinitesimal obstruction to lifting the evolution when the almost complex structure varies. Finally, we construct non-integrable almost-SKT structures with nowhere vanishing Nijenhuis tensor and non-flat characteristic connection on compact six-manifolds, including $S^1\times S^2\times S^3$ and $S^3\times S^3$. In these families, the metric is fixed and the compatible almost complex structures form an $S^2$-family. These structures have vanishing Bismut Ricci form and are therefore almost-CYT.

We extend the classification of manifolds with positive isotropic curvature to all dimensions $n \geq 5$. This result also completes the classification of uniformly positive isotropic curvature on complete non-compact $\kappa$-noncollapsed ancient solutions and gradient shrinking Ricci solitons. The proof involves a construction of pinching cones in all dimensions, refining and extending to all dimensions the results of the first author. We have used MATHEMATICA to find viable parameters for the cone families. The final presentation is self-contained and can be verified manually. No stage of this work involved LLMs.

We study deformations of foliated nonsingular varieties defined by locally free subsheaves of tangent sheaves and locally free subshseaves of cotangent sheaves in the language of functors of Artin rings. We identify first-order deformations and obstructions.

The Ahlfors–Weill construction explicitly extends a univalent function $f$ with Schwarzian norm less than $1/2$ to a quasiconformal homeomorphism of the Riemann sphere, an extension that can be understood geometrically through Epstein surfaces. For asymptotically conformal curves, the uniformization map can be extended to a larger disk, and Loewner theory can be used to verify that this extension is quasiconformal. In this paper, we construct a local quasireflection with such an extension. Using this local quasireflection, we modify classical estimates of Ahlfors and show that domains bounded by asymptotically conformal curves have asymptotically similar hyperbolic and quasihyperbolic metrics. Furthermore, by combining these estimates with a geometric reinterpretation of the extension through Epstein surfaces, we show that the Epstein-Poincaré surfaces of the complementary domains are arbitrarily close to one another near the curve.