Real Analytic Surfaces Research Articles

Geometry of hyperbolic Cauchy–Riemann singularities and KAM-like theory for holomorphic involutions

This article is concerned with the geometry of germs of real analytic surfaces in $$({\mathbb {C}}^2,0)$$ having an isolated Cauchy–Riemann (CR) singularity at the origin. These are perturbations of Bishop quadrics. There are two kinds of CR singularities stable under perturbation: elliptic and hyperbolic. Elliptic case was studied by Moser–Webster (Acta Math 150(3–4), 255–296, 1983) who showed that such a surface is locally, near the CR singularity, holomorphically equivalent to normal form from which lots of geometric features can be read off. In this article we focus on perturbations of hyperbolic quadrics. As was shown by Moser and Webster (1983), such a surface can be transformed to a formal normal form by a formal change of coordinates that may not be holomorphic in any neighborhood of the origin. Given a non-degenerate real analytic surface M in $$({\mathbb {C}}^2,0)$$ having a hyperbolic CR singularity at the origin, we prove the existence of a non-constant Whitney smooth family of connected holomorphic curves intersecting M along holomorphic hyperbolas. This is the very first result concerning hyperbolic CR singularity not equivalent to quadrics. This is a consequence of a non-standard KAM-like theorem for pair of germs of holomorphic involutions $$\{\tau _1,\tau _2\}$$ at the origin, a common fixed point. We show that such a pair has large amount of invariant analytic sets biholomorphic to $$\{z_1z_2=const\}$$ (which is not a torus) in a neighborhood of the origin, and that they are conjugate to restrictions of linear maps on such invariant sets.

Multiplicity, regularity and blow-spherical equivalence of real analytic sets

This article is devoted to studying multiplicity and regularity of analytic sets. We present an equivalence for analytic sets, named blow-spherical equivalence, which generalizes differential equivalence and subanalytic bi-Lipschitz equivalence and, with this approach, we obtain several applications on analytic sets. On multiplicity, we present a generalization for Gau–Lipman’s Theorem about differential invariance of the multiplicity in the complex and real cases, and we show that the multiplicity \(\mathrm{mod}\,2\) is invariant under blow-spherical homeomorphisms in the case of real analytic curves and surfaces and also for a class of real analytic foliations and is invariant by (image) arc-analytic blow-spherical homeomorphisms in the case of real analytic hypersurfaces, generalizing some results proved by G. Valette. On regularity, we show that blow-spherical regularity of real analytic sets implies \(C^1\) smoothness only in the case of real analytic curves. We present also a complete classification of the germs of real analytic curves.

Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group

Abstract We address the problem of integrability of the sub-Riemannian mean curvature of an embedded hypersurface around isolated characteristic points. The main contribution of this paper is the introduction of a concept of a mildly degenerate characteristic point for a smooth surface of the Heisenberg group, in a neighborhood of which the sub-Riemannian mean curvature is integrable (with respect to the perimeter measure induced by the Euclidean structure). As a consequence, we partially answer to a question posed by Danielli, Garofalo and Nhieu in [D. Danielli, N. Garofalo and D. M. Nhieu, Integrability of the sub-Riemannian mean curvature of surfaces in the Heisenberg group, Proc. Amer. Math. Soc. 140 2012, 3, 811–821], proving that the mean curvature of a real-analytic surface with discrete characteristic set is locally integrable.

Symmetries of Einstein-Weyl manifolds with boundary

Starting from a real analytic surface ℳ with a real analytic conformal Cartan connection A. Bor´owka constructed a minitwistor space of an asymptotically hyperbolic Einstein–Weyl manifold with ℳ being the boundary. In this article, starting from a symmetry of conformal Cartan connection, we prove that symmetries of conformal Cartan connection on ℳ can be extended to symmetries of the obtained Einstein–Weyl manifold.

Monomialization of singular metrics on real surfaces

We present here a result of Monomialization of real analytic two-symmetric tensor fields over regular real analytic surfaces. We apply it to the (extension of the pull-back of the) inner metric of a resolved surface of a real analytic surface singularity. Doing so we recover Hsiang & Pati property at each point of the resolved surface.

Real analytic complete non-compact surfaces in Euclidean space with finite total curvature arising as solutions to ODEs

We use the solution space of a pair of ODEs of at least second order to construct a smooth surface in Euclidean space. We describe when this surface is a proper embedding which is geodesically complete with finite total Gauss curvature. If the associated roots of the ODEs are real and distinct, we give a universal upper bound for the total Gauss curvature of the surface which depends only on the orders of the ODEs and we show that the total Gauss curvature of the surface vanishes if the ODEs are second order. We examine when the surfaces are asymptotically minimal.

Focal points and sup-norms of eigenfunctions II: the two-dimensional case

We use a purely dynamical argument on circle maps to improve a result in our accompanying article, [5], on real analytic surfaces possessing eigenfunctions that achieve maximal sup norm bounds. The improved result is that there exists a ‘pole’ p so that all geodesics emanating from p are smoothly closed.

Nodal Sets of Schrödinger Eigenfunctions in Forbidden Regions

This note concerns the nodal sets of eigenfunctions of semiclassical Schrodinger operators acting on compact, smooth, Riemannian manifolds, with no boundary. In the case of real analytic surfaces, we obtain sharp upper bounds for the number of intersections of the zero sets of Schrodinger eigenfunctions with a fixed curve that lies inside the classically forbidden region.

Twistor construction of asymptotically hyperbolic Einstein–Weyl spaces

Starting from a real analytic conformal Cartan connection on a real analytic surface S, we construct a complex surface T containing a family of pairs of projective lines. Using the structure on S we also construct a complex 3-space Z, such that Z is a twistor space of a self-dual conformal 4-fold and T is a quotient of Z by a holomorphic local C× action. We prove that T is a minitwistor space of an asymptotically hyperbolic Einstein–Weyl space with S as an asymptotic boundary.

Twistor holomorphic affine surfaces and projective invariants

We study affine immersions with twistor lifts. Using a decomposition of a connection, we obtain several projective invariants for such affine immersions. In particular, affine immersions with holomorphic twistor lifts are considered. We can show the property that an affine immersion has holomorphic twistor lifts is invariant under projective transformations and characterize immersions with holomorphic twistor lifts by vanishing of some of projective invariants. In the case of compact affine surfaces with holomorphic twistor lifts, we see a quantization phenomenon for one of the projective invariants which we obtain. Moreover, we prove that a real analytic twistor holomorphic affine surface with the symmetric Ricci tensor with respect to both complex structures is totally geodesic or totally umbilic.

Ergodicity and intersections of nodal sets and geodesics on real analytic surfaces

We consider the intersections of the complex nodal set $\mathcal{N}_{\lambda_{j}}^{\,\mathbb{C}}$ of the analytic continuation of an eigenfunction of $\Delta$ on a real analytic surface $(M^2, g)$ with the complexification of a geodesic $\gamma$. We prove that if the geodesic flow is ergodic and if $\gamma$ is periodic and satisfies a generic asymmetry condition, then the intersection points $\mathcal{N}_{\lambda_{j}}^{\,\mathbb{C}} \cap \gamma_{x, \xi}^{\mathbb{C}}$ condense along the real geodesic and become uniformly distributed with respect to its arc-length. We prove an analogous result for non-periodic geodesics except that the ‘origin’ $\gamma_{x, \xi}(0)$ is allowed to move with $\lambda_j$.

QUASIHOMOGENEOUS ANALYTIC AFFINE CONNECTIONS ON SURFACES

We classify torsion-free real-analytic affine connections on compact oriented real-analytic surfaces which are locally homogeneous on a nontrivial open set, without being locally homogeneous on all of the surface. In particular, we prove that such connections exist. This classification relies on a local result that classifies germs of torsion-free real-analytic affine connections on a neighborhood of the origin in the plane which are quasihomogeneous, in the sense that they are locally homogeneous on an open set containing the origin in its closure, but not locally homogeneous in the neighborhood of the origin.

The Nullstellensatz for real coherent analytic surfaces

In this paper we prove Hilbert Nullstellensatz for real coherent analytic surfaces and we give a precise description of the obstruction to get it in general. Refering the first, we prove that the ideals of global functions vanishing on analytic subsets are exactly the real saturated ones. For \mathbb{R}^3 we prove that the real Nullstellensatz holds for real saturated ideals if and only if no principal ideal generated by a function whose zero set is a curve (indeed, a special function) is real. This led us to compare the Nullstellensatz problem with the Hilbert 17th one, also in its weaker form involving infinite sums of squares, proving that they share in fact the same obstruction.

A Bishop surface with a vanishing Bishop invariant

We give a solution to the equivalence problem for Bishop surfaces with the Bishop invariant λ=0. As a consequence, we answer, in the negative, a problem that Moser asked in 1985 after his work with Webster in 1983 and his own work in 1985. This will be done in two major steps: We first derive the formal normal form for such surfaces. We then show that two real analytic Bishop surfaces with λ=0 are holomorphically equivalent if and only if they have the same formal normal form (up to a trivial rotation). Our normal form is constructed by an induction procedure through a completely new weighting system from what is used in the literature. Our convergence proof is done through a new hyperbolic geometry associated with the surface.

Jacobi’s last geometric statement extends to a wider class of Liouville surfaces

Numerical evidence is presented which strongly suggests that “Jacobi’s last geometric statement"—that the conjugate locus from a point has exactly four cusps and the corresponding cut locus consists of only one topological segment—holds for compact real analytic Liouville surfaces diffeomorphic to S 2 S^2 if the Gaussian curvature is everywhere positive and has exactly six critical points, these being two saddles, two global minima, and two global maxima (as is the case for an ellipsoid). Our experiments suggest that this is a sufficient rather than a necessary condition. Furthermore, for compact real analytic Liouville surfaces diffeomorphic to S 2 S^2 upon which the Gaussian curvature can be negative but has exactly six critical points, these being two saddles, two global minima, and two global maxima, it appears that the cut locus is always a subarc of a line given by x 1 = c o n s t x_1=\mathrm {const} or x 2 = c o n s t x_2=\mathrm {const} , where ( x 1 , x 2 ) (x_1,x_2) are canonical coordinates with respect to which the metric has the form ( f 1 ( x 1 ) + f 2 ( x 2 ) ) ( d x 1 2 + d x 2 2 ) (f_1(x_1)+f_2(x_2))(dx_1^2+dx_2^2) . In the case of an ellipsoid, these curves are lines of curvature.

On the Pythagoras numbers of real analytic set germs

We Show that (i) the Pythagoras number of a real analytic set germ is the supremum of the Pythagoras numbers of the curve germs it contains, and (ii) every real analytic curve germ is contained in a real analytic surface germ with the same Pythagoras number (or Pythagoras number 2 if the curve is Pythagorean). This gives new examples and counterexamples concerning sums of squares and positive semidefinite analytic function germs.

The behavior of the principal distributions on a real-analytic surface

The purposes of this paper are: (a) to study the behavior of the principal distributions around an isolated umbilical point on a real-analytic surface of a certain type, which contains all the real-analytic, special Weingarten surfaces;(b) to present one condition such that if a real-analytic surface satisfies the condition, then the index of an isolated umbilical point on the surface is less than or equal to one.

Existence of real analytic surfaces with hyperbolic complex tangent that are formally but not holomorphically equivalent to quadrics

We shall prove the existence of a real analytic sur- face in C 2 that is formally but not holomorphically equivalent to a quadratic surface with a hyperbolic complex tangent. The result is obtained through the pair of holomorphic involutions of Moser-Webster for a real analytic surface with a complex tan- gent of non-vanishing Bishop invariant. Our method is also used to show the existence of real analytic reversible maps of the real plane, defined near the origin, that are formally, but not real an- alytically, equivalent to a linear rotation.

The Artin-Lang property for normal real analytic surfaces

We solve the 17th Hilbert Problem and prove the Artin-Lang property for normal real analytic surfaces. Then we deduce that the absolute (resp. relative) holomorphy ring of such a surface consists of all bounded (resp. locally bounded) meromorphic functions.

A class of real-analytic surfaces in the 3-Euclidean space

A smooth surface $S$ in $\bf{R}^3$ is called parallel curved if there exists a plane in $\bf{R}^3$ such that at each point of $S$, there exists a principal direction parallel to the plane. For example, a plane, a cylinder and a round sphere are parallel curved. More generally, a surface of revolution is also parallel curved. The purposes of this paper are to study the behavior of the principal distributions on a real-analytic, parallel curved surface and to classify the connected, complete, real-analytic, embedded, parallel curved surfaces.