238 research outputs found
Optimal eigenvalue estimate for the Dirac-Witten operator on bounded domains with smooth boundary
Eigenvalue estimate for the Dirac-Witten operator is given on bounded domains
(with smooth boundary) of spacelike hypersurfaces satisfying the dominant
energy condition, under four natural boundary conditions (MIT, APS, modified
APS, and chiral conditions). This result is a generalisation of Friedrich's
inequality for the usual Dirac operator. The limiting cases are also
investigated.Comment: 2007, 18 pages, submitted 02 June 200
Rigidity of compact Riemannian spin Manifolds with Boundary
In this article, we prove new rigidity results for compact Riemannian spin
manifolds with boundary whose scalar curvature is bounded from below by a
non-positive constant. In particular, we obtain generalizations of a result of
Hang-Wang \cite{hangwang1} based on a conjecture of Schroeder and Strake
\cite{schroeder}.Comment: English version of "G\'eom\'etrie spinorielle extrins\`eque et
rigidit\'es", Corollary 6 in Section 3 added, to appear in Letters Math. Phy
Optimal eigenvalues estimate for the Dirac operator on domains with boundary
We give a lower bound for the eigenvalues of the Dirac operator on a compact
domain of a Riemannian spin manifold under the \MIT bag boundary condition.
The limiting case is characterized by the existence of an imaginary Killing
spinor.Comment: 10 page
The Dirac operator on untrapped surfaces
We establish a sharp extrinsic lower bound for the first eigenvalue of the
Dirac operator of an untrapped surface in initial data sets without apparent
horizon in terms of the norm of its mean curvature vector. The equality case
leads to rigidity results for the constraint equations with spherical boundary
as well as uniqueness results for constant mean curvature surfaces in Minkowski
space.Comment: 16 page
On a spin conformal invariant on manifolds with boundary
On a n-dimensional connected compact manifold with non-empty boundary
equipped with a Riemannian metric, a spin structure and a chirality operator,
we study some properties of a spin conformal invariant defined from the first
eigenvalue of the Dirac operator under the chiral bag boundary condition. More
precisely, we show that we can derive a spinorial analogue of Aubin's
inequality.Comment: 26 page
Nonexistence of Generalized Apparent Horizons in Minkowski Space
We establish a Positive Mass Theorem for initial data sets of the Einstein
equations having generalized trapped surface boundary. In particular we answer
a question posed by R. Wald concerning the existence of generalized apparent
horizons in Minkowski space
A Reilly formula and eigenvalue estimates for differential forms
We derive a Reilly-type formula for differential p-forms on a compact
manifold with boundary and apply it to give a sharp lower bound of the spectrum
of the Hodge Laplacian acting on differential forms of an embedded hypersurface
of a Riemannian manifold. The equality case of our inequality gives rise to a
number of rigidity results, when the geometry of the boundary has special
properties and the domain is non-negatively curved. Finally we also obtain, as
a by-product of our calculations, an upper bound of the first eigenvalue of the
Hodge Laplacian when the ambient manifold supports non-trivial parallel forms.Comment: 22 page
Branson's Q-curvature in Riemannian and Spin Geometry
On a closed n-dimensional manifold, n ≥ 5, we compare the three basic conformally covariant operators: the Paneitz-Branson, the Yamabe and the Dirac operator (if the manifold is spin) through their first eigenvalues. On a closed 4-dimensional Riemannian manifold, we give a lower bound for the square of the first eigenvalue of the Yamabe operator in terms of the total Branson's Q-curvature. As a consequence, if the manifold is spin, we relate the first eigenvalue of the Dirac operator to the total Branson's Q-curvature. Equality cases are also characterized
Surgery and the spinorial tau-invariant
We associate to a compact spin manifold M a real-valued invariant \tau(M) by
taking the supremum over all conformal classes over the infimum inside each
conformal class of the first positive Dirac eigenvalue, normalized to volume 1.
This invariant is a spinorial analogue of Schoen's -constant, also
known as the smooth Yamabe number. We prove that if N is obtained from M by
surgery of codimension at least 2, then with . Various topological conclusions
can be drawn, in particular that \tau is a spin-bordism invariant below
. Below , the values of cannot accumulate from
above when varied over all manifolds of a fixed dimension.Comment: to appear in CPD
Contemporary Trends in the Use and Outcomes of Surgical Treatment of Tricuspid Regurgitation
Background
Tricuspid regurgitation (TR), if untreated, is associated with an adverse impact on long‐term outcomes. In recent years, there has been an increasing enthusiasm about surgical and transcatheter treatment of patients with severe TR. We aim to evaluate the contemporary trends in the use and outcomes of tricuspid valve (TV) surgery for TR using the National Inpatient Sample. Methods and Results
Between January 1, 2003 and December 31, 2014, an estimated 45 477 patients underwent TVsurgery for TR in the United States, of whom 15% had isolated TV surgery and 85% had TVsurgery concomitant with other cardiac surgery. There was a temporal upward trend to treat sicker patients during the study period. Patients who underwent isolated TV repair or replacement had a distinctly different clinical risk profile than those patients who underwent TVsurgery simultaneous with other surgery. Isolated TV replacement was associated with high in‐hospital mortality (10.9%) and high rates of permanent pacemaker implantation (34.1%) and acute kidney injury requiring dialysis (5.5%). Similarly, isolated TV repair was also associated with high in‐hospital mortality (8.1%) and significant rates of permanent pacemaker implantation (10.9%) and new dialysis (4.4%). Isolated TV repair and TV replacement were both associated with protracted hospitalizations and substantial cost. Conclusions
In contemporary practice, surgical treatment of TR remains underused and is associated with high operative morbidity and mortality, prolonged hospitalizations, and considerable cost
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