arXiv subjects

Spyros Alexakis

Publications and source records attributed to Spyros Alexakis.

32 records · Page 2Linked to original sources

arXiv2014

On the Geometry of Null Cones to Infinity Under Curvature Flux Bounds

The main objective of this paper is to control the geometry of a future outgoing truncated null cone extending smoothly toward infinity in an Einstein-vacuum spacetime. In particular, we wish to do this under minimal regularity assumptions, namely, at the (weighted) L^2-curvature level. We show that if the curvature flux and the data on an initial sphere of the cone are sufficiently close to the corresponding values in a standard Minkowski or Schwarzschild null cone, then we can obtain quantitative bounds on the geometry of the entire infinite cone. The same bounds also imply the existence of limits at infinity of the natural geometric quantities. Furthermore, we make no global assumptions on the spacetime, as all assumptions are applied only to this single truncated cone. In our sequel paper, we will apply these results in order to control the Bondi energy and the angular momentum associated with this cone.

math.AP↗

arXiv2014

Unique continuation from infinity for linear waves

We prove various uniqueness results from null infinity, for linear waves on asymptotically flat space-times. Assuming vanishing of the solution to infinite order on suitable parts of future and past null infinities, we derive that the solution must vanish in an open set in the interior. We find that the parts of infinity where we must impose a vanishing condition depend strongly on the background geometry. In particular, for backgrounds with positive mass (such as Schwarzschild or Kerr), the required assumptions are much weaker than the ones in the Minkowski space-time. The results are nearly optimal in many respects. They can be considered analogues of uniqueness from infinity results for second order elliptic operators. This work is partly motivated by questions in general relativity.

math.AP↗

arXiv2013

Complete Willmore surfaces in H3 with bounded energy: boundary regularity and bubbling

We study various aspects related to boundary regularity of complete properly embedded Willmore surfaces in H3, particularly those related to assumptions on boundedness or smallness of a certain weighted version of the Willmore energy. We prove, in particular, that small energy controls C1 boundary regularity. We examine the possible lack of C1 convergence for sequences of surfaces with bounded Willmore energy and find that the mechanism responsible for this is a bubbling phenomenon, where energy escapes to infinity.

math.DG↗

arXiv2013

Singular Ricci solitons and their stability under the Ricci flow

We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions $n+1\ge 3$, and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sharp contrast to the evolution of their smooth counterparts. In particular, the family of diffeomorphisms associated with the Ricci flow "pushes away" from the singularity causing the evolving soliton to open up immediately becoming an incomplete (but non-singular) metric. In the second part of this paper we study the local-in time stability of this dynamical evolution, under spherically symmetric perturbations of the initial soliton metric. We prove a local well-posedness result for the Ricci flow near the singular initial data, which in particular implies that the "opening up" of the singularity persists for the perturbations also.

math.AP↗

arXiv2011

The Decomposition of Global Conformal Invariants: Some Technical Proofs. I

This paper forms part of a larger work where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of "global conformal invariants"; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand.

math.DG↗

arXiv2009

The decomposition of Global Conformal Invariants I: On a conjecture of Deser and Schwimmer

This is the first in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global confor- mal invariants"; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. In this paper we set up an iterative procedure that proves the decom- position. We then derive the iterative step in the first of two cases, subject to a purely algebraic result which is proven in [6, 7, 8].

math.DG↗

arXiv2009

The decomposition of global conformal invariants II: The Fefferman-Graham ambient metric and the nature of the decomposition

This is the second in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. The present paper addresses the hardest challenge in this series: It shows how to {\it separate} the local conformal invariant from the divergence term in the integrand; we make full use of the Fefferman-Graham ambient metric to construct the necessary local conformal invariants, as well as all the author's prior work [1,2,3] to construct the necessary divergences. This result combined with [3] completes the proof of the conjecture, subject to establishing a purely algebraic result which is proven in [6,7,8].

math.DG↗

arXiv2009

The decomposition of global conformal invariants IV: A proposition on local Riemannian invariants

This is the fourth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. The present paper lays out the second half of this entire work: The second half proves certain purely algebraic statements regarding local Riemannian invariants; these were used extensively in [3,4]. These results may be of independent interest, applicable to related problems.

math.DG↗

arXiv2009

The decomposition of Global Conformal Invariants V

This is the fifth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. The present paper complements [6] in reducing the purely algebraic results that were used in [3,4 to certain simpler Lemmas, which will be proven in the last paper in this series, [8].

math.DG↗

arXiv2009

The decomposition of global conformal invariants VI: The proof of the proposition on local Riemannian invariants

This is the last in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. The present paper, jointly with [6,7] gives a proof of an algebraic Proposition regarding local Riemannian invariants, which lies at the heart of our resolution of the Deser-Schwimmer conjecture. This algebraic Propositon may be of independent interest, applicable to related problems.

math.DG↗

arXiv2009

Unique continuation for the vacuum Einstein equations

We derive a unique continuation theorem for the vacuum Einstein equations. Our method of proof utilizes Carleman estimates (most importantly one obtained recently by Ionescu and Klainerman), but also relies strongly on certain geometric gauge constructions which make it possible to address this problem via such estimates. We indicate how our method can be used more broadly to derive unique continuation for Einstein's equations from Carleman estimates for the wave operator.

gr-qc↗

arXiv2007

On conformally invariant differential operators

We construct new families of conformally invariant differential operators acting on densities. We introduce a simple, direct approach which shows that all such operators arise via this construction when the degree is bounded by the dimension. The method relies on a study of well-known transformation laws and on Weyl's theory regarding identities holding ``formally'' vs. ``by substitution''. We also illustrate how this new method can strengthen existing results in the parabolic invariant theory for conformal geometries.

math.DG↗

arXiv2006

On the decomposition of global conformal invariants I

This is the first of two papers where we address and partially confirm a conjecture of Deser and Schwimmer, originally postulated in high energy physics. The objects of study are scalar Riemannian quantities constructed out of the curvature and its covariant derivatives, whose integrals over compact manifolds are invariant under conformal changes of the underlying metric. Our main conclusion is that each such quantity that locally depends only on the curvature tensor (without covariant derivatives) can be written as a linear combination of the Chern-Gauss-Bonnet integrand and a scalar conformal invariant.

math.DG↗

arXiv2006

On the decomposition of global conformal invariants II

This paper is a continuation of [2], where we complete our partial proof of the Deser-Schwimmer conjecture on the structure of ``global conformal invariants''. Our theorem deals with such invariants P(g^n) that locally depend only on the curvature tensor R_{ijkl} (without covariant derivatives). In [2] we developed a powerful tool, the ``super divergence formula'' which applies to any Riemannian operator that always integrates to zero on compact manifolds. In particular, it applies to the operator I_{g^n}(ϕ) that measures the ``non-conformally invariant part'' of P(g^n). This paper resolves the problem of using this information we have obtained on the structure of I_{g^n}(ϕ) to understand the structure of P(g^n).

math.DG↗