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Peter Ebenfelt

Publications and source records attributed to Peter Ebenfelt.

At least 37 records · Page 2Linked to original sources
arXiv2016

The Log term in the Bergman and Szeg\H o kernels in strictly pseudoconvex domains in $\mathbb C^2$

In this paper, we consider bounded strictly pseudoconvex domains $D\subset \mathbb C^2$ with smooth boundary $M=M^3:=\partial D$. If we consider the asymptotic expansion of the Bergman kernel on the diagonal $$ K_B\sim \frac{ϕ_B}{ρ^{n+1}}+ψ_B\logρ, $$ where $ρ>0$ is a Fefferman defining equation for $D$, then it is well known that the trace of the log term $bψ_B:=(ψ_B)|_M$ on $M$ does not determine the CR geometry of $M$ locally; e.g., the vanishing of $bψ_B$ on an open subset of $M$ does not imply that $M$ is locally spherical there. Nevertheless, the main result in this paper is that if $D\subset \mathbb C^2$ is assumed to have transverse symmetry, then the global vanishing of $bψ_B$ on $M$ implies that $M$ is locally spherical. A similar result is proved for the Szeg\H o kernel.

math.CV↗
arXiv2016

Invariants and Umbilical Points on Three Dimensional CR Manifolds embedded in $\mathbb C^2$

We introduce a new sequence of CR invariant determinants on a three dimensional CR manifold $M$ embedded in $\mathbb C^2$. The lowest order invariant $\det A_3$ represents E. Cartan's 6th order invariant (the umbilical "tensor"), whose zero locus yields the set of umbilical points on $M$. As an application of this new presentation of the umbilical invariant, we show that generic, almost circular perturbations of the unit sphere always contain curves or surfaces of umbilical points.

math.CV↗
arXiv2015

A normal form for 1-infinite type hypersurfaces in $\mathbb C^2$. I. Formal Theory

In this paper, we study the real hypersurfaces $M$ in $\mathbb C^2$ at points $p\in M$ of infinite type. The degeneracy of $M$ at $p$ is assumed to be the least possible, namely such that the Levi form vanishes to first order in the CR transversal direction. A new phenomenon, compared to known normal forms in other cases, is the presence of resonances as roots of an universal polynomial in the $7$-jet of the defining function of $M$. The main result is a complete (formal) normal form at points $p$ with no resonances. Remarkably, our normal form at such infinite type points resembles closely the Chern-Moser normal form at Levi-nondegenerate points. For a fixed hypersurface, its normal forms are parametrized by $S^1\times \mathbb R^*$, and as a corollary we find that the automorphisms in the stability group of $M$ at $p$ without resonances are determined by their $1$-jets at $p$. In the last section, as a contrast, we also give examples of hypersurfaces with arbitrarily high resonances that possess families of distinct automorphisms whose jets agree up to the resonant order.

math.CV↗
arXiv2015

On the HJY Gap Conjecture in CR geometry vs. the SOS Conjecture for polynomials

We show that the Huang-Ji-Yin (HJY) Gap Conjecture concerning CR mappings between spheres follows from a conjecture regarding Sums of Squares (SOS) of polynomials. The connection between the two problems is made by the CR Gauss equation and the fact that the former conjecture follows from the latter follows from a recent result, due to the author, on partial rigidity of CR mappings of strictly pseudoconvex hypersurfaces into spheres.

math.CV↗
arXiv2015

Umbilical points on three dimensional strictly pseudoconvex CR manifolds. I. Manifolds with $U(1)$-action

The question of existence of umbilical points, in the CR sense, on compact, three dimensional, strictly pseudoconvex CR manifolds was raised in the seminal paper by S.-S. Chern and J. K. Moser in 1974. In the present paper, we consider compact, three dimensional, strictly pseudoconvex CR manifolds that possess a free, transverse action by the circle group $U(1)$. We show that every such CR manifold $M$ has at least one orbit of umbilical points, {\it provided} that the Riemann surface $X:=M/U(1)$ is not a torus. In particular, every compact, circular and strictly pseudoconvex hypersurface in $\mathbb C^2$ has at least one circle of umbilical points. The existence of umbilical points in the case where $X$ is a torus is left open in general, but it is shown that if such an $M$ has additional symmetries, in a certain sense, then it must possess umbilical points as well.

math.CV↗
arXiv2014

Local Holomorphic Isometries of a Modified Projective Space into a Standard Projective Space; Rational Conformal Factors

We consider local modifications $ω_n+f^*ω_d$ of the Fubini-Study metric (with associated $(1,1)$-form $ω_n$) on an open subset $Ω\subset \bC\bP^n$ induced by a local holomorphic mapping $f\colon Ω\to \bP^d$. Our main result is that there are "gaps" in potential dimensions $m$ such that the modification can be obtained as $h^*ω_m$ for some local holomorphic mapping $h\colon Ω\to \bC\bP^m$. We also consider the case of rational conformal factors.

math.CV↗
arXiv2013

Proper holomorphic mappings into $\ell$-concave quadric domains in projective space

In this paper, we prove a type of partial rigidity result for proper holomorphic mappings of certain $\ell$-concave domains in projective space into model quadratic $\ell$-concave domains. The main technical result is a degree estimate for proper holomorphic mappings into the model domains, provided that the mappings extend to projective space as rational mappings, and the source domain contains algebraic varieties and has a boundary with low CR complexity.

math.CV↗
arXiv2013

Extension of Solutions to Holomorphic Partial Differential Equations

Given a strictly pseudoconvex domain G and a linear partial differential operator P with holomorphic coefficients, we derive sufficient conditions for the existence of a solution to Pu = 0 which is holomorphic in G near a point p in the boundary or G, but cannot be extended holomorphically across p.

math.AP↗
arXiv2012

Partial rigidity of degenerate CR embeddings into spheres

In this paper, we study degenerate CR embeddings $f$ of a strictly pseudoconvex hypersurface $M\subset \bC^{n+1}$ into a sphere $\bS$ in a higher dimensional complex space $\bC^{N+1}$. The degeneracy of the mapping $f$ will be characterized in terms of the ranks of the CR second fundamental form and its covariant derivatives. In 2004, the author, together with X. Huang and D. Zaitsev, established a rigidity result for CR embeddings $f$ into spheres in low codimensions. A key step in the proof of this result was to show that degenerate mappings are necessarily contained a complex plane section of the target sphere (partial rigidity). In the 2004 paper, it was shown that if the total rank $d$ of the second fundamental form and all of its covariant derivatives is $<n$ (here, $n$ is the CR dimension of $M$), then $f(M)$ is contained in a complex plane of dimension $n+d+1$. The converse of this statement is also true, as is easy to see. When the rank $d$ exceeds $n$, it is no longer true, in general, that $f(M)$ is contained in a complex plane of dimension $n+d+1$, as can be seen by examples. In this paper, we carry out a systematic study of degenerate CR mappings into spheres. We show that when the ranks of the second fundamental form and its covariant derivatives exceed the CR dimension $n$, then partial rigidity may still persist, but there is a "defect" $k$ that arises from the ranks exceeding $n$ such that $f(M)$ is only contained in a complex plane of dimension $n+d+k+1$. Moreover, this defect occurs in general, as is illustrated by examples.

math.CV↗
arXiv2012

On CR embeddings of strictly pseudoconvex hypersurfaces into spheres in low dimensions

It follows from the 2004 work of the first author, X.Huang, and D. Zaitsev that any local CR embedding $f$ of a strictly psedoconvex hypersurface $M^{2n+1}\subset\bC^{n+1}$ into the sphere $\bS^{2N+1}\subset \bC^{N+1}$ is rigid, i.e.\ any other such local embedding is obtained from $f$ by composition by an automorphism of the target sphere $\bS^{2N+1}$, {\it provided} that the codimension $N-n<n/2$. In this paper, we consider the limit case $N-n=n/2$ in the simplest situation where $n=2$, i.e.\ we consider local CR embeddings $f\colon M^5\to \bS^7$. We show that there are at most two different local embeddings, up to composition with an automorphism of $\bS^7$. We also identify a subclass of 5-dimensional, strictly pseudoconvex hypersurfaces $M^5$ in terms of their CR curvatures such that rigidity holds for local CR embeddings $f\colon M^5\to \bS^7$.

math.CV↗
arXiv2011

Transversality of holomorphic mappings between real hypersurfaces in complex spaces of different dimensions

We consider holomorphic mappings $H$ between a smooth real hypersurface $M\subset \bC^{n+1}$ and another $M'\subset \bC^{N+1}$ with $N\geq n$. We provide conditions guaranteeing that $H$ is transversal to $M'$ along all of $M$. In the strictly pseudoconvex case, this is well known and follows from the classical Hopf boundary lemma. In the equidimensional case ($N=n$), transversality holds for maps of full generic rank provided that the source is of finite type in view of recent results by the authors (see also a previous paper by the first author and L. Rothschild). In the positive codimensional case ($N>n$), the situation is more delicate as examples readily show. In recent work by S. Baouendi, the first author, and L. Rothschild, conditions were given guaranteeing that the map $H$ is transversal outside a proper subvariety of $M$, and examples were given showing that transversality may fail at certain points. One of the results in this paper implies that if $N\le 2n-2$, $M'$ is Levi-nondegenerate, and $H$ has maximal rank outside a complex subvariety of codimension 2, then $H$ is transversal to $M'$ at all points of $M$. We show by examples that this conclusion fails in general if $N\geq 2n$, or if the set $W_H$ of points where $H$ is not of maximal rank has codimension one. We also show that $H$ is transversal at all points if $H$ is assumed to be a finite map (which allows $W_H$ to have codimension one) and the stronger inequality $N\leq 2n-3$ holds, provided that $M$ is of finite type.

math.CV↗
arXiv2010

Partial Rigidity of CR Embeddings of Real Hypersurfaces into Hyperquadrics with Small Signature Difference

We study the rigidity of holomorphic mappings from a neighborhood of a Levi-nondegenerate CR hypersurface $M$ with signature $l$ into a hyperquadric $Q_{l'}^{N} \subseteq \mathbb{CP}^{N+1}$ of larger dimension and signature. We show that if the CR complexity of $M$ is not too large then the image of $M$ under any such mapping is contained in a complex plane with dimension independent of $N$. This result follows from two theorems, the first demonstrating that for sufficiently degenerate mappings, the image of $M$ is contained in a plane, and the second relating the degeneracy of mappings into different quadrics.

math.CV↗
arXiv2010

CR transversality of holomorphic mappings between generic submanifolds in complex spaces

We show that a holomorphic mapping sending one generic submanifold into another of the same dimension is CR transversal to the target submanifold provided that the source manifold is of finite type and the map is of generic full rank. This result and its corollaries completely resolve two questions posed by Linda P. Rothschild and the first author in a paper from 2006.

math.CV↗
arXiv2009

Holomorphic Mappings between Hyperquadrics with Small Signature Difference

In this paper, we study holomorphic mappings sending a hyperquadric of signature $\ell$ in $\bC^n$ into a hyperquadric of signature $\ell'$ in $\bC^N$. We show (Theorem \ref{main}) that if the signature difference $\ell'-\ell$ is not too large, then the mapping can be normalized by automorphisms of the target hyperquadric to a particularly simple form and, in particular, the image of the mapping is contained in a complex plane of a dimension that depends only on $\ell$ and $\ell'$, and not on the target dimension $N$. We also prove a Hopf Lemma type result (Theorem \ref{main2}) for such mappings.

math.CV↗
arXiv2007

The Goursat problem for a generalized Helmholz operator in the plane

We consider the Goursat problem in the plane for partial differential operators whose principal part is the $p$th power of the standard Laplace operator. The data is posed on a union of $2p$ distinct lines through the origin. We show that the solvability of this Goursat problem depends on Diophantine properties of the geometry of lines on which the data is posed.

math.AP↗
arXiv2007

On the mixed Cauchy problem with data on singular conics

We consider a problem of mixed Cauchy type for certain holomorphic partial differential operators whose principal part $Q_{2p}(D)$ essentially is the (complex) Laplace operator to a power, $Δ^p$. We pose inital data on a singular conic divisor given by P=0, where $P$ is a homogeneous polynomial of degree $2p$. We show that this problem is uniquely solvable if the polynomial $P$ is elliptic, in a certain sense, with respect to the principal part $Q_{2p}(D)$.

math.AP↗