Sketch of Wickramasekera’s Regularity Theorem
Suppose \(V_i\) is a sequence of stationary integral \(n\)-varifolds in \(B_2^{n+1}(0)\) and \(V_i\) also satisfies the following conditions:
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\(0\in \mathrm{spt}\|V_i\|\).
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\(\|V_i\|(B_2^{n+1}(0))\leq \Lambda\) for some constant \(\Lambda\gt{}0\) independent of \(i\).
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(Stability) Each \(V_i\) is stable in \(B_2^{n+1}(0)\) on its regular set, i.e., for any \(\phi \in C_c^1(\mathrm{reg}V_i)\),
- (Alpha-Structural Hypothesis) There exists \(\alpha\in (0,1)\) such that for each \(i\), no point of \(\mathrm{spt}\|V_i\|\cap B_1^{n+1}(0)\) has a neighborhood in which \(\mathrm{spt}\|V_i\|\) is the union of three or more embedded \(C^{1,\alpha}\) hypersurfaces-with-boundary meeting only along their common boundary.
Then, up to a subsequence, \(V_i\) converges in the varifold sense to a stationary integral \(n\)-varifold \(V_\infty\) in \(B_2^{n+1}(0)\), which is stable and whose singular set in \(B_2^{n+1}(0)\) has Hausdorff dimension at most \(n-7\).
We write the class of such varifolds as \(\mathcal{V}_\alpha(\Lambda)\).
Let \(V_i\in \mathcal{V}_\alpha(\Lambda)\) such that \(V_i \to q|B_2^n(0) \times \{ 0 \}\) in the varifold sense. Then, for \(i\) large enough, we have
with \(u_{i,1}\leq \cdots \leq u_{i,q}\) and \(u_{i,k}\) are smooth functions such that
for some constant \(C=C(n,\alpha,\Lambda)\gt{}0\) independent of \(i\).
Suppose \(\boldsymbol{C}\) is a classical cone. There is no sequence of varifolds \(V_i\in \mathcal{V}_\alpha(\Lambda)\) such that \(V_i \to \boldsymbol{C}|_{B_2(0)}\) in the varifold sense.
Recall that a classical cone
where \(H_i\) are \(n\)-dimensional half-hyperplanes in \(\mathbb{R}^{n+1}\) such that they contains origin and share the same boundary and \(q_i\) are positive integers.
Sketch of the proof
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The two theorems are proved simultaneously by induction on the density of planes and cones. Assume that the sheeting theorem holds for \(q\leq q_0\) and that the minimal distance theorem holds for \(\Theta(\boldsymbol{C},0)\leq q_0\).
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Show that minimal distance theorem holds for \(\Theta(\boldsymbol{C},0)\leq q_0+1\). It is enough to treat the densities \(\Theta(\boldsymbol{C},0)=q_0+\frac{1}{2}\) and \(q_0+1\).
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Show that the sheeting theorem holds for \(q=q_0+1\).
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Dimension reduction argument plus the classification of stable cones implies that the singular set of \(V_\infty\) has Hausdorff dimension at most \(n-7\).
Proof of the minimal distance theorem
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Assume the sheeting theorem holds for \(q\leq q_0\) and the minimal distance theorem holds for \(\Theta(\boldsymbol{C},0)\leq q_0\). Together with dimension reduction, this implies the desired regularity whenever \(\Theta(\|V\|,X)\leq q_0\) for all \(X\in B_2(0)\).
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Suppose \(V\) is sufficiently close to a classical cone \(\boldsymbol{C}\) in the varifold sense and \(\Theta(\|V\|,0)\geq \Theta(\|\boldsymbol{C}\|,0)\). Then, outside neighborhood of the singular set of \(\boldsymbol{C}\), using the sheeting theorem, we can write \(V\) as a union of smooth graphs over the half-hyperplanes in \(\boldsymbol{C}\).
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Establish the \(L^{2}\)-estimate for these graphs and key \(L^{2}\) improvement in a smaller scale, following Simon’s cylindrical-singularity argument [Sim93].
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By iteration, we can obtain the set \(\{ \Theta(\|V\|,X)\geq \Theta(\|\boldsymbol{C}\|,0)\}\) is \(C^{1,\alpha}\)-regular for some \(\alpha\gt{}0\), which implies \(V\) has classical singularity at \(0\), which is a contradiction.
Proof of the sheeting theorem
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Same induction assumption as above.
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Write \(V_i\) as a union of Lipschitz graphs \(u_{i,k}\) over the plane \(\{ x_{n+1}=0 \}\) outside a small bad set.
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Consider the blow-up class \(v\), which is the limit of \(E(V_i)^{-1}u_{i,k}\) where \(E(V_i)\) is the \(L^{2}\)-excess of \(V_i\).
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(\(\star\)) Prove that \(v=(v_k)\) is harmonic.
First, we need a Lipschitz approximation for stationary varifolds.
Fix \(q\in\mathbb{Z}_{\geq 1}\) and \(\sigma\in(0,1)\). Then there exists \(\varepsilon_0=\varepsilon_0(n,q,\sigma)\in(0,1)\) such that the following holds. Let \(V\) be a stationary integral \(n\)-varifold in \(B_2^{n+1}(0)\) such that:
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\(\displaystyle \frac{1}{\omega_n 2^n}\,\|V\|\bigl(B_2^{n+1}(0)\bigr) \lt{} q+\frac12\) and \(\displaystyle q-\frac12 \leq \frac{1}{\omega_n}\,\|V\|\bigl(\mathbb{R}\times B_1^n(0)\bigr) \lt{} q+\frac12\);
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\(\displaystyle \hat{E}^2(V) := \int_{\mathbb{R}\times B_1^n(0)} |x^1|^2\,\mathrm{d}\|V\|(X) \lt{} \varepsilon_0\).
Then there exists \(\mathcal{H}^n\)-measurable \(\Sigma\subset B_\sigma^n(0)\) such that:
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\(\displaystyle \mathcal{H}^n(\Sigma)+\|V\|(\mathbb{R}\times\Sigma) \leq C\,\hat{E}_V^2\);
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there exist Lipschitz functions \(u_1,\ldots,u_q:B_\sigma^n(0)\to\mathbb{R}\) with \(\operatorname{Lip}(u_j)\leq\frac12\), \(u_1\leq\cdots\leq u_q\), \(\displaystyle \sup_{B_\sigma^n(0)} |u_j| \leq C\,\hat{E}_V^{\frac{1}{n+1}}\), and
Here \(C=C(n,q,\sigma)\in(0,\infty)\).
Moreover, we can establish the following estimate
We consider \(V_i\) as above such that \(\hat{E}^{2}(V_i) \to 0^+\) and \(\sigma_i \to 1\) as \(i\to \infty\). The above estimate implies that \(\hat{u}_k:=E^{-1}u_k\) satisfies
Up to a subsequence, we can find a weak limit function \(v\). The blow-up class \(\mathcal{B}_q\) is the collection of all such limits.
The blow-up class \(\mathcal{B}_q\) has the following properties:
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\(\mathcal{B}_q\subset W^{1,2}_{\mathrm{loc}}(B_1^n(0);\mathbb{R}^q)\cap L^2(B_1^n(0);\mathbb{R}^q)\).
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If \(v\in\mathcal{B}_q\), then \(v^1\leq v^2\leq\cdots\leq v^q\).
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If \(v\in\mathcal{B}_q\), then \(\Delta v_a=0\) in \(B_1^n(0)\), where \(v_a=q^{-1}\sum_{j=1}^q v^j\).
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For each \(v\in\mathcal{B}_q\) and each \(z\in B_1^n(0)\), either (\(\mathcal{B}\)4 I**)** or (\(\mathcal{B}\)4 II**)** below holds:
(\(\mathcal{B}\)4 I**)** (Hardt–Simon inequality [HS79]). For each \(\rho\in\bigl(0,\tfrac{3}{8}(1-|z|)\bigr]\),
where \(R_z(x)=|x-z|\), \(\ell_{v,z}(x)=v_a(z)+Dv_a(z)\cdot(x-z)\), and \(v-\ell_{v,z}=\bigl(v^1-\ell_{v,z},\ldots,v^q-\ell_{v,z}\bigr)\).
**(**<span class="math inline">\(\mathcal{B}\)</span><!-- -->4 *II***)** There exists <span class="math inline">\(\sigma=\sigma(z)\in(0,1-|z|]\)</span> such that <span class="math inline">\(\Delta v=0\)</span> in <span class="math inline">\(B_\sigma^n(z)\)</span>.
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If \(v\in\mathcal{B}_q\), then:
(\(\mathcal{B}\)5 I**)** \(\displaystyle \tilde{v}_{z,\sigma}(\cdot)\equiv \|v(z+\sigma(\cdot))\|_{L^2(B_1^n(0))}^{-1}\,v(z+\sigma(\cdot))\in\mathcal{B}_q\) for each \(z\in B_1^n(0)\) and \(\sigma\in\bigl(0,\tfrac{3}{8}(1-|z|)\bigr]\) whenever \(v\not\equiv 0\) in \(B_\sigma^n(z)\);
(\(\mathcal{B}\)5 II**)** \(v\circ\gamma\in\mathcal{B}_q\) for each orthogonal rotation \(\gamma\) of \(\mathbb{R}^n\);
(\(\mathcal{B}\)5 III**)** \(\|v-\ell_v\|_{L^2(B_1^n(0))}^{-1}(v-\ell_v)\in\mathcal{B}_q\) whenever \(v-\ell_v\not\equiv 0\) in \(B_1^n(0)\), where \(\ell_v(x)=v_a(0)+Dv_a(0)\cdot x\) for \(x\in\mathbb{R}^n\) and \(v-\ell_v=\bigl(v^1-\ell_v,\ldots,v^q-\ell_v\bigr)\).
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If \(\{v_k\}_{k=1}^\infty\subset\mathcal{B}_q\), then there exists a subsequence \(\{k'\}\) of \(\{k\}\) and a function \(v\in\mathcal{B}_q\) such that \(v_{k'}\to v\) locally in \(L^2(B_1^n(0))\) and locally weakly in \(W^{1,2}(B_1^n(0))\).
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If \(v\in\mathcal{B}_q\) is such that for each \(j=1,\ldots,q\), there exist linear maps \(L_1^j,L_2^j:\mathbb{R}^n\to\mathbb{R}\) with \(v^j(x^2,y)=L_1^j(x^2,y)\) if \(x^2\gt{}0\), \(v^j(x^2,y)=L_2^j(x^2,y)\) if \(x^2\leq 0\), and \(L_1^j(0,y)=L_2^k(0,y)\) for all \(1\leq j,k\leq q\) and \(y\in\mathbb{R}^{n-1}\), where \((x^2,y)\) are coordinates on \(\mathbb{R}^n\), then \(v^1=v^2=\cdots=v^q=L\) for some linear map \(L:\mathbb{R}^n\to\mathbb{R}\).
Proof.
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This a consequence of the definition of the blow-up class.
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This is based on \(u_k\leq u_{k+1}\).
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This is from the stationary property of \(V_i\).
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We can consider a new sequence \(V_i'=(\eta_{z,\sigma})_{\#}V_i\), where its blow-up limit is \(\tilde{v}_{z,\sigma}\). Similarly, we can consider \(V_i'=\gamma_{\#}V_i\), where its blow-up limit is \(v \circ \gamma\). Finally, the last property is from the rotation involving direction \(e_{n+1}\).
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This follows from the compactness of the blow-up sequence.
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This is one of the key properties of the blow-up class, and the place that stable condition is used. The idea is, either we have good density points accumulating at a given point, which implies the Hardt–Simon inequality. Or we have a density gap, which by our induction assumption, implies that \(v\) is harmonic.
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This is the most technical part of the proof. It rules out the possibility that a blow-up class contains the singular model of a classical cone.
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For any \(v\in\mathcal{B}_q\), we have \(v\) is smooth and harmonic.
Proof. This proof is similar to the classical dimensional reduction argument.
We define the “singular set” of \(v\) as the set of points \(x\in B_1^n(0)\) such that \(\mathcal{B}4 \mathrm{I}\) holds.
First, we can study homogeneous degree-one functions in the blow-up class, namely tangent functions of a given function.
Claim. Each homogeneous 1 function in the blow-up class is a linear function.
Next, we can show that the “singular set” is empty by the dimensional reduction argument. ◻
After proving the blow-up class are all smooth and harmonic, we can show the sheeting theorem holds by using the standard regularity argument.
Proof of the property \(\mathcal{B}7\)
Suppose \(v\in\mathcal{B}_q\) is such that for each \(j=1,\ldots,q\), we can write
for some \(\lambda^j,\mu^j\in\mathbb{R}\).
We define the cone \(\boldsymbol{C}_k\) as the union of the graphs of the functions \(\hat{E}_i \lambda^j x_1\) for \(x_1\gt{}0\) and \(\hat{E}_i \mu^j x_1\) for \(x_1\leq 0\).
We can define the fine excess \(E(V_i)\) as follows:
Clearly, we have \(E(V_i) \leq \hat{E}(V_i)\). But we expect that \(E(V_i)\) is much smaller than \(\hat{E}(V_i)\).
Indeed, we can establish the following under suitable assumptions.
By the sheeting theorem, we can write \(V_i\) as a union of smooth graphs over the cones \(\boldsymbol{C}_i\) for \(|x_1|\geq \sigma\).
Now, we define \(h_i\) and \(w_i\) the vector valued function such that
and we expect that \(h_i^j\) and \(w_i^j\) much smaller than \(\hat{E}_i\).
Establish the key \(L^{2}\)-estimate for \(h_i^j,w_i^j\)
Such estimate containst the following
We can also establish the uniform bound for \(E^{-1}(V_i)h_i^j\) and \(E^{-1}(V_i)w_i^j\) and obtain the limit function \(h\) and \(w\). Such limit is called the fine blow-up limit.
For the fine blow-up limit \(h\) and \(w\), we know \(h,w\) are at least \(C^{2}\) up to the boundary.
Proof. This relies on the key \(L^{2}\) esimate using the fine excess.
First, we need to show the \(C^{0,\alpha}\) estimate for \(h,w\). We need to use the \(L^{2}\)-estimate to finish the proof.
Now, we need to show the \(C^{2,\alpha}\) estimate for \(h,w\). This is based on the stationary property of \(V_i\) and the definition of the blow-ups, and the properties of harmonic functions.
Note that each \(h,w\) are all harmonic in its interior, which is a consequence of the stationary property of \(V_i\) and definition of the blow-ups.
The key is actually the smoothness up to the boundary. ◻
Improvement of the fine excess at a smaller scale
Using the properties of the fine blow-up limit, we can improve the fine excess at a smaller scale as follows.
Under suitable assumptions, with \(\hat{E}(V_i)\) small enough, \(\hat{E}^{-1}(V_i)E(V_i)\) is small enough, we can find a new cone \(\boldsymbol{C}_i'\), such that if we apply a rotation, a scaling to the original varifold \(V_i\), we can obtain a new varifold \(V_i'\), such that the fine excess of \(V_i'\) with respect to \(\boldsymbol{C}_i'\) is much smaller than the fine excess of \(V_i\) with respect to \(\boldsymbol{C}_i\). One can think of this as follows. Under suitable assumptions, if one zooms in the original varifold \(V_i\), it will look closer to a new cone \(\boldsymbol{C}_i'\).
Iterative process
If such steps can be done infinitely many times for a given varifold \(V_i\), we can obtain a limit cone \(\boldsymbol{C}_{i,\infty}\). Note that the difference between \(\boldsymbol{C}_{i,k}\) and \(\boldsymbol{C}_{i,k+1}\) is much smaller than the difference between \(\boldsymbol{C}_{i,k}\) and the plane \(\{ x_{n+1}=0 \}\), because the fine excess is much smaller than the original excess. Hence \(\boldsymbol{C}_{i,\infty}\) is still a classical cone. These steps also imply that we can find varifolds \(V_{i,k}\) which converge to \(\boldsymbol{C}_{i,\infty}\) in the varifold sense.
This is a contradiction to the minimal distance theorem.
Gaoming Wang
Assistant Professor
My research interests include Geometric Analysis and Partial Differential Equations.