Codimension-Two Bulk Extremal Surfaces

• Codimension-two bulk extremal surfaces are spacelike submanifolds in Lorentzian spaces that extremize area and other functionals.
• They use n_r-Gauss maps to provide a rigorous framework for analyzing extrinsic curvatures, including flatness and umbilicity conditions.
• These surfaces are pivotal in mathematical relativity and holography, aiding in the classification of geometric structures and variational problems.

A codimension-two bulk extremal surface is a (typically spacelike) submanifold of codimension two that extremizes a geometric functional—most commonly area—in a Lorentzian or pseudo-Riemannian ambient space. Such surfaces play a pivotal role in Lorentzian geometry, the mathematics of submanifold theory, and modern theoretical physics, especially in contexts where extrinsic curvature, notions of umbilicity, and geometric classification of surfaces are required.

1. Geometric Construction in Lorentz–Minkowski Space

The systematic paper of spacelike codimension-two surfaces in Lorentz–Minkowski space $\mathbb{R}^{n+1}_1$ requires tools that go beyond classical Gauss maps, which are naturally tied to codimension-one (hypersurface) settings. In this context, the notion of an $HS_r$-valued Gauss map provides a canonical method for assigning normal data to each point on such a surface (1102.2527).

Given a spacelike surface $M$ with codimension two, the normal plane $N_p M$ at each $p \in M$ is a timelike 2-plane. Fixing a base point $v = (0, \ldots, 0, -1)$, one considers the model hyperbolic space of radius one,

$$H^n(v,1) = \{x \in \mathbb{R}^{n+1}_1 : (x-v, x-v) = -1,\, x_{n+1} \geq 0\}$$

where $(\cdot,\cdot)$ denotes the Lorentzian dot product. Translated normal planes (passing through the origin in this frame) intersect this hyperbolic space in a hyperbola.

For each fixed $r > 0$, introduce the hyperplane $\Pi_r = \{x \in \mathbb{R}^{n+1}_1 : x_{n+1} = r\}$. The intersection $HS_r := H^n(v,1) \cap \Pi_r$ consists of two points for each $p$, denoted $\mathbf{n}_r^+(p)$ and $\mathbf{n}_r^-(p)$. The pair of maps

$\mathbf{n}_r^+,\, \mathbf{n}_r^- : M \to HS_r$

are then called the $n_r^{\pm}$-Gauss maps. These maps provide a replacement for the unit normal in the higher-codimension setting and are shown to be smooth and well-defined via a local implicit function theorem applied to a specific system of algebraic constraints.

2. Differential Geometry and Fundamental Curvatures

Once defined, the $n_r^{\pm}$-Gauss maps yield a powerful framework for analyzing the extrinsic geometry of $M$. Their differentials decompose into tangential and normal components:

$$d\mathbf{n}_r(p) = (d\mathbf{n}_r)^{\top}(p) + (d\mathbf{n}_r)^{N}(p)$$

The associated Weingarten map is

$$A^{n_r} = - (d\mathbf{n}_r)^{\top}: T_p M \to T_p M$$

which is symmetric. At $p$, its eigenvalues, denoted $k_1^{n_r}(p),\dotsc, k_{n-1}^{n_r}(p)$, serve as the $n_r$-principal curvatures. The corresponding curvatures are:

• $n_r$-Gauss–Kronecker curvature: $K_p^{n_r} = \det(A^{n_r})$
• $n_r$-mean curvature: $$H_p^{n_r} = \frac{1}{n-1}\operatorname{tr}(A^{n_r})$$

A point $p$ is called $n_r$-umbilic if all $n_r$-principal curvatures coincide, and $n_r$-flat if they all vanish. When $M$ is $n_r$-umbilic (respectively, $n_r$-flat) at every point for every $r > 0$, it is termed totally umbilic (resp., totally flat).

3. Characterization of Flat and Umbilic Surfaces

The $n_r^{\pm}$-Gauss maps offer analytic criteria linking the constancy and geometric properties of these maps to the global structure of $M$ (1102.2527):

• $M$ is $n_r$-flat for some $r$ if and only if $\mathbf{n}_r^+$ or $\mathbf{n}_r^-$ is constant. This is equivalent to $M$ being contained in an affine hyperplane.
• If $M$ is $n_r$-umbilic for some $r$ and subject to appropriate geometric constraints (e.g., lying in a pseudo-hypersphere), it is actually totally umbilic. In particular, umbilicity relative to two independent normal fields (e.g., both $n_r^+$ and $n_r^-$) implies umbilicity relative to any smooth normal field, simplifying classification.

Precise theorems formalize these statements. For instance, in the context of $M$ contained in a hyperbolic space $H^n(0,R)$, the equivalence of $n_r$-umbilicity, total umbilicity, and containment in a hyperplane is established.

4. Codimension-Two Extremal Surfaces and Broader Significance

In higher-codimension Lorentzian geometry and physical theories (notably, general relativity and holography), codimension-two extremal surfaces are integral to the analysis of bulk geometry. They arise naturally when studying:

• Variational problems such as hypersurfaces (minimal or extremal area) in time-oriented spacetimes,
• The Ryu–Takayanagi and Hubeny–Rangamani–Takayanagi formulas for entanglement entropy, where the surfaces in question are codimension-two and must be extremal in the bulk,
• Problems in rigidity, stability, and classification of globally or locally extreme objects under geometric flows or constraints.

The $\mathbf{n}_r^{\pm}$-Gauss maps furnish invariant curvature quantities defined extrinsically via the ambient Lorentzian geometry, facilitating rigorous criteria for flatness and umbilicity that are valid even when classical codimension-one tools are insufficient.

Moreover, since these Gauss maps target a hyperbolic space, they link the paper of such surfaces with other geometric settings where hyperbolic geometry is fundamental, which is common in relativistic and holographic applications.

5. Main Formulas and Computational Framework

A concise summary of the foundational mathematical objects and formulas:

• Model hyperbolic space:

$$H^n(v, 1) = \{ x \in \mathbb{R}^{n+1}_1 : (x - v, x - v) = -1,\, x_{n+1} \geq 0 \}$$

• Intersection hyperplane:

$$\Pi_r = \{ x \in \mathbb{R}^{n+1}_1 : x_{n+1} = r \}$$

• $n_r^{\pm}$-Gauss maps:

$n_r^+, n_r^- : M \to HS_r = H^n(v, 1) \cap \Pi_r$

• Weingarten map and principle curvatures:

$$A^{n_r} = - (dn_r)^{\top}, \quad \text{eigenvalues: } k_i^{n_r}(p)$$

• Curvature invariants:

$$K_p^{n_r} = \det(A^{n_r}), \quad H_p^{n_r} = \frac{1}{n-1} \operatorname{tr}(A^{n_r})$$

In local coordinates, the points $a = (a_1, \dots, a_n, r)$ representing $n_r^{\pm}(p)$ are determined by the linear system: $$\begin{aligned} & (X_{u_i}(p), a - v) = 0 \quad \text{for } i = 1, \dots, n-1, \ & (a - v, a - v) = -1. \end{aligned}$$ Here $X(u)$ is a local parametrization of $M$ and $\{ X_{u_1}, \dots, X_{u_{n-1}} \}$ spans the tangent space.

6. Applications and Contextual Significance

The introduction of $n_r^{\pm}$-Gauss maps enables a complete characterization of flat and umbilic codimension-two surfaces in Lorentz–Minkowski space. This has ramifications for:

• Analytic description and classification of physically significant surfaces in mathematical relativity,
• The paper of bulk extremal surfaces in AdS/CFT and related dualities, especially where higher codimension or nontrivial extrinsic geometry is involved,
• Generalizations to variational problems for area, WiLLMore energy, and related functionals in pseudo-Riemannian settings.

By encoding curvature data in the behavior of $n_r^{\pm}$, one obtains a bridge from local geometric invariants to overarching extrinsic structures (such as totally umbilic or totally flat surfaces) and associated global embedding problems.

7. Summary Table: Core Geometric Structures

Structure	Definition/Construction	Key Role
$H^n(v,1)$	Model hyperbolic space in $\mathbb{R}^{n+1}_1$	Target for Gauss map; ambient normal geometry
$HS_r$	$H^n(v,1) \cap \{x_{n+1}=r\}$	Range of $n_r^{\pm}$-Gauss map
$n_r^{\pm}$-Gauss maps	Intersection points of translated normal and $HS_r$	Encodes extrinsic normals at each $p \in M$
$A^{n_r}$	Tangential derivative of $n_r^{\pm}$, $-(dn_r)^{\top}$	Self-adjoint, yields curvatures
$K_p^{n_r}$, $H_p^{n_r}$	Det(A), Tr(A)/(n-1)	$n_r$-Gauss–Kronecker and mean curvatures

This framework, rooted in the construction of $n_r^{\pm}$-Gauss maps, is essential for the analysis and classification of codimension-two bulk extremal surfaces in Lorentzian geometry and remains widely applicable in mathematical physics, particularly in geometric analysis and relativistic field theories.