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Aaron J. Tyrrell

Publications and source records attributed to Aaron J. Tyrrell.

13 recordsLinked to original sources
arXiv2026

Renormalized Area of Catenoids in the Hyperbolic Space

We show that the generating curves of non-totally geodesic spherical rotational minimal hypersurfaces (catenoids, for simplicity) of the hyperbolic spaces $\mathbb{H}^{2n+1}$ are $p$-elastic curves for $p=(2n-1)/(2n)$. We employ this variational characterization, together with the Chern--Gauss--Bonnet formulas for locally conformally flat manifolds, to present an explicit expression for the renormalized area of catenoids in terms of hyperelliptic integrals. Further analyzing these special integrals, we show that the renormalized area of catenoids varies continuously from negative infinity to twice the renormalized area of the totally geodesic hypersurfaces $\mathbb{H}^{2n}\subset\mathbb{H}^{2n+1}$. Therefore, we conclude that the renormalized area is not bounded below and that, when $n$ is even, the renormalized area of minimal hypersurfaces in $\mathbb{H}^{2n+1}$ does not have a sign.

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arXiv2026

Renormalized area of minimal surfaces in hyperbolic space

In this paper we consider minimal surfaces in hyperbolic space of arbitrary codimension that are critical for renormalized area. We give two criteria which imply that $Y$ must be a totally geodesic disk. One can be viewed as a 'gap' result for the renormalized area.

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arXiv2026

Topological and rigidity results for four-dimensional hypersurfaces in space forms

Exploiting the special features of four-dimensional Riemannian geometry, we derive topological and rigidity results for hypersurfaces immersed in space forms of dimension 5. First, we provide a complete description of the Weyl tensor for four-dimensional hypersurfaces, by means of which we derive a new characterization result for isoparametric hypersurfaces; then, we prove sharp topological bounds on the Weyl functional for closed, minimal hypersurfaces, involving the Euler characteristic in the case of an ambient space with constant non-negative sectional curvature. Then, inspired by a famous conjecture by Chern and the so-called second pinching problem, we find estimates for the norm of the second fundamental form in terms of the Euler characteristic in the minimal, constant scalar curvature case, under a cross-sectional area assumption. Finally, we prove some rigidity results by means of integral inequalities on the derivatives of the second fundamental form, also dealing with special curvature conditions, such as half harmonic Weyl curvature and Bach-flatness. We also extend some of the local results to the case of a locally conformally flat 5-dimensional ambient space.

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arXiv2026

Computing renormalized curvature integrals on Poincaré-Einstein manifolds

We describe a general procedure for computing renormalized curvature integrals on Poincaré-Einstein manifolds. In particular, we explain the connection between the Gauss-Bonnet-type formulas of Albin and Chang-Qing-Yang for the renormalized volume, and explicitly identify a scalar conformal invariant in the latter formula. Our approach constructs scalar conformal invariants of weight $-n$ on $n$-manifolds, $n \geq 8$, that are natural divergences; these imply that the scalar invariant in the Chang-Qing-Yang formula is not unique in dimension $n \geq 8$. Our procedure also produces explicit conformally invariant Gauss--Bonnet-type formulas for compact Einstein manifolds.

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arXiv2026

Local and global conformal invariants of submanifolds

We develop methods for constructing and computing conformal invariants of submanifolds, with a particular emphasis on conformal submanifold scalars and conformally invariant integrals of natural submanifold scalars. These methods include a direct construction of the extrinsic ambient space, a construction of global invariants of conformally compact minimal submanifolds of conformally compact Einstein manifolds via renormalized extrinsic curvature integrals, and the introduction of a large class of conformal submanifold scalars that are easily computed at minimal submanifolds of Einstein manifolds. As an application, we derive an explicit Gauss--Bonnet--Chern-type formula relating the renormalized area of a conformally compact $k$-dimensional minimal submanifold of a conformally compact Einstein manifold to its Euler characteristic and the integral of a conformal submanifold scalar of weight $-k$. As another application, we prove a rigidity result for conformally compact minimal submanifolds of conformally compact hyperbolic manifolds.

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arXiv2025

Holography and Cheeger constant of asymptotically CMC submanifolds

Let $(M^{n+1},g_+)$ be an asymptotically hyperbolic manifold. We compute the Cheeger constant of conformally compact asymptotically constant mean curvature submanifolds $ ι: Y^{k+1} \to (M^{n+1},g_+)$ with arbitrary codimension. As an application, we provide two classes of examples of $(n+1)$-dimensional asymptotically hyperbolic manifolds with Cheeger constant equal to $n$, whose conformal infinity is of the following types: 1) positive Yamabe invariant, and 2) negative Yamabe invariant. Moreover, in the same spirit as Blitz--Gover--Waldron \cite{BlitzSamuel2021CFFa}, we show that an asymptotically hyperbolic manifold with umbilic boundary is conformally weakly Poincaré--Einstein if and only if the third conformal fundamental form of the boundary vanishes. Next, in the space of asymptotically minimal hypersurfaces $Y$ within a Poincaré--Einstein manifold, we identify an extrinsic conformal invariant of $\partial Y$ which obstructs the vanishing of the mean curvature of $Y$ to second order. This conformal invariant is a linear combination of two Riemannian hypersurface invariants of $\partial Y,$ one which depends on its extrinsic geometry within $\overline{Y}$ and the other on its extrinsic geometry within $\partial M;$ neither of which are conformal invariants individually. Finally, we show that for asymptotically minimal hypersurfaces with mean curvature vanishing to second order inside of a Poincaré--Einstein space, being weakly Poincaré--Einstein is equivalent to the boundary of $Y$ having vanishing second and third conformal fundamental forms when viewed as a hypersurface within the conformal infinity.

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arXiv2025

A Gauss-Bonnet formula for the renormalized area of minimal submanifolds of Poincaré-Einstein manifolds

Assuming the extrinsic $Q$-curvature admits a decomposition into the Pfaffian, a scalar conformal submanifold invariant, and a tangential divergence, we prove that the renormalized area of an even-dimensional minimal submanifold of a Poincaré-Einstein manifold can be expressed as a linear combination of its Euler characteristic and the integral of a scalar conformal submanifold invariant. We derive such a decomposition of the extrinsic $Q$-curvature in dimensions two and four, thereby recovering and generalizing results of Alexakis-Mazzeo and Tyrrell, respectively. We also conjecture such a decomposition for general natural submanifold scalars whose integral over compact submanifolds is conformally invariant, and verify our conjecture in dimensions two and four. Our results also apply to the area of a compact even-dimensional minimal submanifold of an Einstein manifold.

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arXiv2024

First Eigenvalue Estimates for Asymptotically Hyperbolic Manifolds and their Submanifolds

We derive a sharp upper bound for the first eigenvalue $λ_{1,p}$ of the $p$-Laplacian on asymptotically hyperbolic manifolds for $1<p<\infty$. We then prove that a particular class of conformally compact submanifolds within asymptotically hyperbolic manifolds are themselves asymptotically hyperbolic. As a corollary, we show that for any minimal conformally compact submanifold $Y^{k+1}$ within $\mathbb{H}^{n+1}(-1)$, $λ_{1,p}(Y)=\left(\frac{k}{p}\right)^{p}$. We then obtain lower bounds on $λ_{1,2}(Y)$ in the case where minimality is replaced with a bounded mean curvature assumption and where the ambient space is a general Poincaré-Einstein space whose conformal infinity is of non-negative Yamabe type. In the process, we introduce an invariant $\hat β^Y$ for each such submanifold, enabling us to generalize a result due to Cheung-Leung.

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arXiv2024

Self-dual and even Poincaré-Einstein metrics in dimension four

We prove rigidity and gap theorems for self-dual and even Poincaré-Einstein metrics in dimension four. As a corollary, we give an obstruction to the existence of self-dual Poincaré-Einstein metrics in terms of conformal invariants of the boundary and the topology of the bulk. As a by-product of our proof we identify a new scalar conformal invariant of three-dimensional Riemannian manifolds.

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arXiv2023

Renormalized Area for Minimal Hypersurfaces of 5D Poincaré-Einstein Spaces

In this paper we derive a Gauss-Bonnet formula for the renormalized area of Graham-Witten minimal hypersurfaces of 5-dimensional Poincaré-Einstein spaces. The formula we derive expresses the renormalized area in terms of integrals of pointwise scalar Riemannian invariants. We also prove a result which gives a characterization of minimal hypersurfaces with $L^2$ second fundamental form in terms of conformal geometry at infinity.

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arXiv2023

A Sharp Inequality for Trace-Free Matrices with Applications to Hypersurfaces

We derive a sharp inequality relating the second and fourth elementary symmetric functions of the eigenvalues of a trace-free matrix and give two applications. First, we give a new proof of the classification of conformally flat hypersurfaces in spaceforms. Second, we construct a functional which characterizes rotational hypersurfaces and catenoids.

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arXiv2023

A Rigidity Result for Minimal Rotation Hypersurfaces of 5D Spaces of Constant Curvature

In this paper we show that a particular extrinsic pointwise hypersurface invariant is always non-positive on minimal hypersurfaces of constant curvature spaces and vanishes identically if and only if the hypersurface is rotational. We show this for hypersurfaces of 5-dimensional spaces of constant curvature but we conjecture that this should generalize to a similar result in other dimensions.

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