Varifolds

Definition 3.1.1

An \(n\)-varifold \(V\) in \(\mathbb{R}^{n+k}\) is a Radon measure on \(\mathbb{R}^{n+k}\times G(n+k,n)\), where \(G(n+k,n)\) is the set of all \(n\)-dimensional subspaces in \(\mathbb{R}^{n+k}\).

Let \(M \subset \mathbb{R}^{n+k}\) be an \(n\)-dimensional manifold, and let \(\theta\) be an \(\mathcal{H}^n\)-measurable function on \(M\). Then the \(n\)-varifold \(V=|(M,\theta)|\) is defined by

\[ V(U)=\int_{(x,T_xM)\in U} \theta(x) d\mathcal{H}^n|_M(x). \]

Equivalently, for any continuous compactly supported function \(f\), we have

\[ V(f)=\int_{ } \theta(x)f(x,T_xM) d \mathcal{H}^n|_{M}(x). \]

Thus, a varifold arising from an \(n\)-dimensional manifold records tangent-plane information as well. If \(\theta=1\), we usually write \(|(M,1)|=|M|\). For example, \(P\) is the tangent plane of \(M\) at \(x\) if and only if the measure \(|M|\) restricted to \(\{ x \} \times G(n+k,n)\) is nonzero.

Definition 3.1.2

The weight measure \(\|V\|\) of an \(n\)-varifold \(V\) is defined by

\[ \|V\|(A)=V(A \times G(n+m,n)), \]

for any Borel subset \(A \subset \mathbb{R}^{n+m}\). Hence, \(\|V\|\) is a Radon measure on \(\mathbb{R}^{n+m}\). The support of \(\|V\|\), denoted by \(\mathrm{spt}\|V\|\), is defined by

\[ \mathrm{spt}\|V\|=\left\{ x \in \mathbb{R}^{n+k}: \|V\|(B^{n+k}_r(x))\gt{}0 \text{ for any }r\gt{}0 \right\}. \]

The (\(n\)-dimensional) density of \(\|V\|\) at \(x\) is defined by

\[ \Theta(\|V\|,x)=\lim_{r\to 0^+}\frac{\|V\|(B^{n+k}_r(x))}{\omega_nr^n}, \]

if the limit exists, where \(\omega_n\) is the volume of the unit ball in \(\mathbb{R}^n\).

Theorem 3.1.3

Suppose \(V_i\) is a sequence of \(n\)-varifolds such that for every compact \(K \subset \mathbb{R}^{n+k}\), there exists \(C=C(K)\) with

\[ V_i(K \times G(n+k,n))\leq C, \]

then, up to a subsequence, we can find \(V_i \to V\) in the sense of Radon measures. (The convergence is in the varifold sense.) Equivalently,

\[ \lim_{i \to \infty}V_i(f)=V(f) \]

for any \(f \in C_c(\mathbb{R}^{n+k}\times G(n+k,n))\).

Suppose \(V_n\) is defined by

\[ \sum_{i=1}^{2^n} |([0,1]\times \{ \frac{i}{2^n} \}, \frac{i}{2^n})|, \]

Then it converges to \(V\) in the varifold sense, where

\[ V(f)=\int_{0}^{1} \int_{0}^{1} f(x,y, \{ x_2=0 \})dx dy. \]

Note that \(\mathrm{spt}\|V\|=[0,1]^{2}\), so \(V\) cannot be written as \(V=|(M,\theta)|\) for any one-dimensional manifold \(M\).

Definition 3.1.4

We say \(M\) is countably \(n\)-rectifiable if \(M \subset N \cup \bigcup_{j=1}^\infty N_j\) where \(\mathcal{H}^n(N)=0\) and each \(N_j\) is an \(n\)-dimensional embedded \(C^1\) submanifold of \(\mathbb{R}^{n+k}\).

Equivalently, \(M\) is countably \(n\)-rectifiable if and only if there exists a countable family of Lipschitz maps \(f_j: \mathbb{R}^n \to \mathbb{R}^{n+k}\) such that

\[ M = N \cup \bigcup_{j=1}^\infty f_j(A_j), \]

where \(A_j \subset \mathbb{R}^n\) and \(\mathcal{H}^n(N)=0\).

For any countably \(n\)-rectifiable set \(M\), we write \(T_xM\) for the approximate tangent space of \(M\).

Definition 3.1.5

Let \(M\) be an \(\mathcal{H}^n\)-measurable subset of \(\mathbb{R}^{n+k}\) with \(\mathcal{H}^n(M\cap K)\lt{}+\infty\) for every compact subset \(K\). We say that an \(n\)-dimensional subspace \(P\) is an approximate tangent space of \(M\) at \(x\) if and only if

\[ \lim_{r\to 0^+} \int_{ \eta_{x,r}(M)}f(y) d\mathcal{H}^n(y) = \int_{P} f(y) d\mathcal{H}^n(y),\quad \text{for any }f \in C_c(\mathbb{R}^{n+k}). \]
Theorem 3.1.6

If \(M\) is rectifiable, then for \(\mathcal{H}^n\)-almost every \(x \in M\), there exists a unique approximate tangent space \(T_xM\) of \(M\) at \(x\).

Definition 3.1.7

We say an \(n\)-varifold \(V\) is rectifiable if there exists a countably \(n\)-rectifiable, \(\mathcal{H}^n\)-measurable subset \(M\) of \(\mathbb{R}^{n+k}\) and a positive locally \(\mathcal{H}^n\)-integrable function \(\theta\) on \(M\) such that

\[ V(f)=\int_{ M} f(x,T_xM)\theta(x)d \mathcal{H}^n(x) \]

We use the notation \(V=|(M,\theta)|\) for the varifold associated with \(M\) and \(\theta\).

Definition 3.1.8

Let \(V\) be an \(n\)-varifold in \(\mathbb{R}^{n+k}\) and let \(F: \mathbb{R}^{n+k} \to \mathbb{R}^{m+l}\) be a \(C^1\) map. The pushforward varifold \(F_{\#}V\) is defined by

\[ F_{\#}V(\phi) = \int_{ } \phi(F(x), DF_x(S)) \left\vert J_F(x, S) \right\vert \, dV(x,S), \]

for any continuous function \(\phi\) with compact support on \(\mathbb{R}^{m+l} \times G(m+l, n)\), where \(J_F(x, S)\) is the Jacobian of \(F\) restricted to \(S\), i.e.,

\[ J_F(x, S) = \sqrt{\det(\frac{\partial F}{\partial \tau_i} \cdot \frac{\partial F}{\partial \tau_j})}, \]

where \(\{ \tau_i \}_{i=1}^n\) is an orthonormal basis of \(S\).

Let \(F_t\) be a one-parameter family of diffeomorphisms on \(\mathbb{R}^{n+k}\) with \(F_0\) being the identity map. The first variation of an \(n\)-varifold \(V\) under the variation \(F_t\) is defined by

\[ \left.\frac{d}{dt}\right|_{t=0} \| (F_t)_{\#} V \|(K), \]

for any compact subset \(K \subset \mathbb{R}^{n+k}\).

Theorem 3.1.9

We have

\[ \left.\frac{d}{dt}\right|_{t=0} \| (F_t)_{\#} V \|(K) = \int_{ } \mathrm{div}^S \varphi(x) dV(x,S). \]
Definition 3.1.10

We define the first variation of \(V\) as a linear functional \(\delta V\) on \(C_c(\mathbb{R}^{n+k}, \mathbb{R}^{n+k})\) by

\[ \delta V(\varphi) = \int_{ } \mathrm{div}^S \varphi(x) dV(x,S). \]
Definition 3.1.11

We say that \(V\) has bounded first variation if \(\delta V\) is a bounded linear functional on \(C_c(\mathbb{R}^{n+k}, \mathbb{R}^{n+k})\).

Hence, by the Riesz representation theorem, there exists a Radon measure \(\|\delta V\|\) and a \(\|\delta V\|\)-measurable vector-valued function \(\nu_V\) such that \(|\nu_V(x)|=1\) for \(\|\delta V\|\)-almost every \(x\) and

\[ \delta V(\varphi) = \int \langle \nu_V(x), \varphi(x) \rangle d\|\delta V\|(x). \]

In particular, we can decompose \(\|\delta V\|=h\|V\|+\sigma_V\) into the absolutely continuous part \(h\|V\|\) and the singular part \(\sigma_V\) with respect to \(\|V\|\).

Let \(\mu\) and \(\nu\) be two Radon measures on \(\mathbb{R}^{n+k}\). We say \(\mu\) is absolutely continuous with respect to \(\nu\) (denoted \(\mu \ll \nu\)) if for every Borel set \(A\), \(\nu(A) = 0\) implies \(\mu(A) = 0\). By the Radon-Nikodym theorem, if \(\mu \ll \nu\), then there exists a \(\nu\)-measurable function \(h\) such that

\[ \mu(A) = \int_A h \, d\nu \]

for all Borel sets \(A\).

Conversely, a measure \(\mu\) is singular with respect to \(\nu\) if there exists a Borel set \(A\) such that \(\nu(A) = 0\) and \(\mu(\mathbb{R}^{n+k} \setminus A) = 0\). Intuitively, \(\mu\) and \(\nu\) are supported on disjoint sets.

Any Radon measure \(\mu\) can be uniquely decomposed as

\[ \mu = \mu_{\mathrm{ac}} + \mu_{\mathrm{sing}}, \]

where \(\mu_{\mathrm{ac}} \ll \nu\) and \(\mu_{\mathrm{sing}} \perp \nu\). So, if \(V\) has bounded first variation, we can write

\[ \delta V(\varphi) =-\int \langle \vec{H}, \varphi(x) \rangle d\|V\|(x) + \int \langle \nu_V(x), \varphi(x) \rangle d\sigma_V(x). \]

where \(\vec{H} =-h \nu_V\) is called the generalized mean curvature vector of \(V\), and \(\nu_V\sigma_V\) is called the generalized boundary of \(V\).

Suppose \(M=\{ (\cos \theta, \sin \theta): \theta \in (0,\pi) \}\), the 1-dimensional half-circle in \(\mathbb{R}^2\), and \(V=|M|\) is the associated varifold. Then, we have

\[ \begin{aligned} \delta V(\varphi)={}&\int_{ M} \mathrm{div}^M \varphi(x) d\mathcal{H}^1|_M(x)=\int_{ M} \mathrm{div}^M \varphi^\top(x) d\mathcal{H}^1|_M(x)+\int_{ M} \varphi(x) \cdot x\\ ={}& \varphi(1,0) \cdot (0,-1) + \varphi(-1,0) \cdot (0,-1) + \int_{ M} \varphi(x) \cdot x d\mathcal{H}^1|_M(x). \end{aligned} \]

So the generalized mean curvature vector \(\vec{H}=-x\) is just the usual mean curvature vector of \(M\), and the generalized boundary \(\nu_V\sigma_V=(0,-1)\delta_{(1,0)} + (0,-1)\delta_{(-1,0)}\) represents the two boundary points of \(M\) with the corresponding outward normal vectors.

Suppose \(M=[0,1]\times \{ 0 \}, \theta(x,y)=x\), and \(V=|(M,\theta)|\) is the associated varifold. Then, we have

\[ \delta V = \int_{ 0}^1 \frac{\partial }{\partial x} \varphi^\top(x,0) x dx = \varphi^\top(1,0)\cdot (1,0) - \int_{ 0}^1 \varphi^\top(x,0)\cdot (1,0) dx. \]

So we have

\[ \vec{H}=(\frac{1}{x},0),\quad \nu_V\sigma_V=(1,0)\delta_{(1,0)}. \]

Thus, the generalized mean curvature vector \(\vec{H}\) depends not only on the geometry of \(M\), but also on the weight function \(\theta\).

This example also shows that, unlike the case of smooth submanifolds, the generalized mean curvature vector of a varifold may not be perpendicular to the tangent plane. If we restrict to integral rectifiable varifolds, we have the following result of Brakke [Bra15].

Theorem 3.1.12

Suppose \(V\) is an integral rectifiable varifold with bounded first variation, then the generalized mean curvature vector \(\vec{H}\) is perpendicular to the tangent plane of \(V\) for \(\|V\|\)-almost every point.

Definition 3.1.13

We say that \(V\) is a stationary varifold in \(U\) if for any \(\varphi \in C_c(U,\mathbb{R}^{n+k})\), we have

\[ \int_{ } \mathrm{div}^S \varphi(x)dV(x,S)=0. \]

If \(V=|M|\), then \(V\) being stationary means

\[ \int_{ M} \mathrm{div}^M \varphi(x) d\mathcal{H}^n|_M(x)=0. \]

This is equivalent to saying that \(M\) is minimal.

This notion of stationarity is very weak. Triple-junctions of three half-planes meeting at \(120\) degrees are stationary.

Theorem 3.1.14

Let \(V\) be a stationary \(n\)-varifold in \(\mathbb{R}^{n+k}\). Then the function

\[ \Phi(r) = \frac{\|V\|(B_r(x_0))}{\omega_n r^n} \]

is monotone increasing in \(r \gt{} 0\). Moreover, we have

\[ \Phi(r)-\Phi(s)=\int_{ B_r(x_0)\backslash B_s(x_0)} \frac{|(y-x_0)^\perp|^2}{|y-x_0|^{n+2}} d\|V\|(y) \]

Proof. We can choose test vector fields \(\varphi\) as before. Suppose \(x_0=0\) for simplicity.

\[ \varphi = \begin{cases} y\left( \frac{1}{|y|^n}-\frac{1}{\rho^n} \right), & \sigma\leq |y|\leq \rho\\ y\left( \frac{1}{\sigma^n}-\frac{1}{\rho^n} \right), & |y|\lt{}\sigma\\ \end{cases} \]

We insert \(\varphi\) into the first variation formula and get the desired monotonicity formula. ◻

One can obtain a modified monotonicity formula for varifolds with \(L^p\)-integrable mean curvature (\(p\gt{}n\)) and no generalized boundary.

Theorem 3.1.15

Suppose \(V_i\) is a sequence of rectifiable stationary varifolds in \(U\) and for any \(K \subset \subset U\), \(\sup\|V_i\|(K)\lt{}+\infty\). We also assume \(\Theta(\|V_i\|,x)\geq 1\) for almost every \(x\in \mathrm{spt}\|V_i\|\). Then, up to a subsequence, there exists a rectifiable stationary varifold \(V\) in \(U\) such that \(V_i\to V\) in the varifold sense.

In particular, if each \(V_i\) is integral, so is \(V\).

  • The condition \(\Theta\geq 1\) is essential.

  • Stationarity is also essential: \(V_n=\sum_{i=1}^{2^n}|[\frac{2i-1}{2^{n+1}}, \frac{2i}{2^{n+1}}]|\) converges to \(|([0,1],\frac{1}{2})|\).

Definition 3.1.16

The tangent cone of an \(n\)-varifold \(V\) at \(x\), denoted by \(\mathrm{VarTan}(V,x)\), is defined by

\[ \mathrm{VarTan}(V,x):= \{V':V'=\lim_{i\to \infty}(\eta_{x,\rho_i})_\#V\text{ for some }\rho_i\to 0^+ \}. \]

The tangent cone at infinity, denoted by \(\operatorname{VarTan}(V,\infty)\), is defined by

\[ \mathrm{VarTan}(V,\infty):= \{V':V'=\lim_{i\to \infty}(\eta_{0,\rho_i})_\#V\text{ for some }\rho_i\to +\infty\}. \]

The tangent cone of a varifold may not be unique. Right now, we do not even know if the tangent cone of any stationary rectifiable varifold is unique or not. This is still an open problem.

Proposition 3.1.17

If \(V\) is stationary in \(\mathbb{R}^{n+k}\), then the tangent cone of \(V\) at \(\infty\) or \(x\) is a stationary cone.

Finally, for any \(n\)-varifold defined on \(U\), \(\mathrm{reg}\|V\|\) denotes the regular set of \(\mathrm{spt}\|V\|\) in \(U\), i.e., the set of points \(x \in \mathrm{spt}\|V\|\) such that there exists \(r\gt{}0\) with \(B^{n+1}_r(x)\subset U\) and \(\mathrm{spt}\|V\|\cap B^{n+1}_r(x)\) is a smooth (immersed) hypersurface in \(B^{n+1}_r(x)\). \(\mathrm{sing}\|V\|\) denotes the singular set of \(\mathrm{spt}\|V\|\) in \(U\).

Allard’s regularity theorem is a foundational result in geometric measure theory. It gives conditions under which a stationary varifold is regular (i.e., smooth) near a point.

Theorem 3.1.18

Let \(V\) be an \(n\)-dimensional stationary integral rectifiable varifold in \(B_2(0)\subset \mathbb{R}^{n+k}\), and suppose that

\[ \|V\|(B_2(0))\leq (1+\delta)\omega_n 2^n \]

for some \(0\lt{}\delta\lt{}1\). Then, there exists \(\varepsilon=\varepsilon(n,\delta)\) such that if

\[ E^{2}:=\int_{ B_2(0)} \operatorname{dist}^2(x,P)\,d\|V\|(x)\leq \varepsilon \]

where \(P=\{ x_{n+1}=0 \}\), then there exists a function \(u \in C^{1,\alpha}(\bar{B_1^n(0)})\) such that

\[ \mathrm{spt}\|V\| \cap B_1(0) = \{ (x',u(x')): x'\in B_1^n(0) \} \]

and

\[ \sup_{B_1^n(0)} |u| + \|D u\|_{L^{\infty}(B_1^n(0))} + [D u]_{C^{0,\alpha}(B_1^n(0))} \leq C(n,\delta) E. \]

Note that \(\delta\) cannot be 1 since otherwise, we have the scaled catenoid as a counterexample.

Theorem 3.1.19

Let \(V\) be an \(n\)-dimensional stationary integral rectifiable varifold in an open set \(U \subset \mathbb{R}^{n+k}\). Suppose that the density of \(\|V\|\) at a point \(x_0 \in U\) is \(1\). Then \(x_0\in \mathrm{reg}\|V\|\).

Gaoming Wang
Gaoming Wang
Assistant Professor

My research interests include Geometric Analysis and Partial Differential Equations.