[proofplan] We blow up the varifold around $x_0$ by scales $r_j \downarrow 0$ and use the monotonicity formula to obtain uniform local mass bounds for the rescaled varifolds. Allard compactness then gives a subsequential integral-varifold limit on every compact subset of $\mathbb{R}^n$. Stationarity passes to the limit by continuity of first variation under varifold convergence. Finally, the density ratios of the blow-ups converge to the density of $V$ at $x_0$, so the limit has constant density ratio; the equality case in the monotonicity formula forces the limit to be invariant under all dilations, hence conical. [/proofplan]

[step:Choose shrinking scales and define the blow-up varifolds] Let $(r_j)_{j=1}^{\infty}$ be any sequence of positive [real numbers](/page/Real%20Numbers) such that $r_j \downarrow 0$ and $B(x_0,r_j)\subset U$ for every $j$. Define the affine rescaling map \begin{align*} \eta_{x_0,r_j}: U \to \mathbb{R}^n \end{align*} by \begin{align*} \eta_{x_0,r_j}(x)=\frac{x-x_0}{r_j}. \end{align*} Let \begin{align*} V_j := (\eta_{x_0,r_j})_{\#}V \end{align*} denote the pushforward $m$-varifold. Its domain is \begin{align*} U_j := \eta_{x_0,r_j}(U)=\frac{U-x_0}{r_j}. \end{align*} Since $r_j \downarrow 0$ and $U$ is an open neighbourhood of $x_0$, every compact set $K\subset \mathbb{R}^n$ is contained in $U_j$ for all sufficiently large $j$. [/step]

[step:Use monotonicity to obtain uniform mass bounds on fixed balls]Fix $R>0$. For all sufficiently large $j$, $B(x_0,r_jR)\subset U$. By the scaling rule for the mass measure of a pushforward $m$-varifold under the dilation $\eta_{x_0,r_j}$, \begin{align*} \|V_j\|(B(0,R)) = r_j^{-m}\|V\|(B(x_0,r_jR)). \end{align*} Using the definition of the density ratio, this becomes \begin{align*} \|V_j\|(B(0,R)) = \omega_m R^m \Theta(V,x_0,r_jR). \end{align*} Since $V$ is stationary in $U$, the monotonicity formula for stationary varifolds applies (citing a result not yet in the wiki: monotonicity formula for stationary varifolds). Hence the function \begin{align*} \rho \mapsto \Theta(V,x_0,\rho) \end{align*} is nondecreasing for $0<\rho<\operatorname{dist}(x_0,\partial U)$. Because the density $\Theta(V,x_0)$ exists and is finite for an integral varifold, the values $\Theta(V,x_0,r_jR)$ remain bounded for each fixed $R$. Therefore there is a constant \begin{align*} M_R := \omega_m R^m \sup_{j \ge j_R}\Theta(V,x_0,r_jR)<\infty \end{align*} such that \begin{align*} \|V_j\|(B(0,R)) \le M_R \end{align*} for all $j\ge j_R$, where $j_R\in \mathbb{N}$ is chosen so that $B(x_0,r_jR)\subset U$ for all $j\ge j_R$.[/step]

[guided]The point of this step is to produce the compactness hypothesis needed for a varifold limit. We fix a radius $R>0$ in the blown-up coordinates and compare the mass of $V_j$ in $B(0,R)$ with the mass of the original varifold $V$ in the small ball $B(x_0,r_jR)$. The rescaling map is the function \begin{align*} \eta_{x_0,r_j}: U \to \mathbb{R}^n \end{align*} defined by \begin{align*} \eta_{x_0,r_j}(x)=\frac{x-x_0}{r_j}. \end{align*} It sends $B(x_0,r_jR)$ exactly onto $B(0,R)$. Since $V$ is $m$-dimensional, mass scales by the factor $r_j^{-m}$ under this dilation. Thus \begin{align*} \|V_j\|(B(0,R)) = r_j^{-m}\|V\|(B(x_0,r_jR)). \end{align*} Now insert the definition of the density ratio: \begin{align*} \Theta(V,x_0,r_jR) = \frac{\|V\|(B(x_0,r_jR))}{\omega_m(r_jR)^m}. \end{align*} Solving this identity for $\|V\|(B(x_0,r_jR))$ and substituting gives \begin{align*} \|V_j\|(B(0,R)) = r_j^{-m}\omega_m(r_jR)^m\Theta(V,x_0,r_jR) = \omega_m R^m \Theta(V,x_0,r_jR). \end{align*} We now use stationarity. The monotonicity formula for stationary varifolds says that \begin{align*} \rho \mapsto \Theta(V,x_0,\rho) \end{align*} is nondecreasing on the interval where $B(x_0,\rho)\subset U$ (citing a result not yet in the wiki: monotonicity formula for stationary varifolds). Because $V$ is integral, the density \begin{align*} \Theta(V,x_0)=\lim_{\rho\downarrow 0}\Theta(V,x_0,\rho) \end{align*} exists and is finite. Therefore the sequence $\Theta(V,x_0,r_jR)$ is bounded for each fixed $R>0$. Defining \begin{align*} M_R := \omega_m R^m \sup_{j \ge j_R}\Theta(V,x_0,r_jR), \end{align*} where $j_R$ is chosen so that $B(x_0,r_jR)\subset U$ for $j\ge j_R$, we obtain \begin{align*} \|V_j\|(B(0,R)) \le M_R. \end{align*} This is exactly the uniform local mass bound required for varifold compactness.[/guided]

[step:Extract an integral varifold limit on compact subsets]For each $R>0$, the preceding step gives a uniform bound for $\|V_j\|(B(0,R))$ along the tail of the sequence. We also verify the first-variation hypothesis in Allard compactness. Let \begin{align*} X: B(0,R) \to \mathbb{R}^n \end{align*} be a compactly supported smooth vector field, extended by zero to $\mathbb{R}^n$. For all sufficiently large $j$, $\operatorname{spt}X\subset U_j$. Define \begin{align*} Y_j: U \to \mathbb{R}^n \end{align*} by \begin{align*} Y_j(x)=r_jX(\eta_{x_0,r_j}(x)). \end{align*} Then $Y_j$ is a compactly supported smooth vector field in $U$, and the chain rule for first variation under the affine dilation gives \begin{align*} \delta V_j(X)=r_j^{-m}\delta V(Y_j)=0, \end{align*} because $V$ is stationary in $U$. Thus the first variations of the $V_j$ are locally uniformly bounded, in fact identically zero on each fixed compact set for all large $j$. Since each $V_j$ is an integral $m$-varifold, the local mass bounds and the verified local first-variation bounds allow Allard compactness to apply on every ball $B(0,R)$ (citing a result not yet in the wiki: Allard [compactness theorem](/theorems/2748) for integral varifolds, including closure of integral varifolds under locally bounded first variation). Using a diagonal subsequence over the exhaustion $B(0,k)$, $k\in\mathbb{N}$, we obtain a subsequence, still denoted $(V_j)$, and an integral $m$-varifold $C$ in $\mathbb{R}^n$ such that \begin{align*} V_j \to C \end{align*} as varifolds on compact subsets of $\mathbb{R}^n$.[/step]

[guided]Allard compactness for integral varifolds requires more than local mass bounds when we want the limit to remain integral: it also requires a local bound on the first variation measures. The mass bound was proved in the previous step, so we now check the first-variation condition directly from stationarity. Fix $R>0$ and let \begin{align*} X: B(0,R) \to \mathbb{R}^n \end{align*} be a compactly supported smooth vector field, extended by zero to $\mathbb{R}^n$. Since $U_j$ exhausts $\mathbb{R}^n$, the support of $X$ lies in $U_j$ for all sufficiently large $j$. For such $j$, define the pulled-back test field \begin{align*} Y_j: U \to \mathbb{R}^n \end{align*} by \begin{align*} Y_j(x)=r_jX(\eta_{x_0,r_j}(x)). \end{align*} This is the correct scaling because $D\eta_{x_0,r_j}=r_j^{-1}I_n$, so the factor $r_j$ cancels the derivative of the dilation when computing tangential divergence. More explicitly, for each $m$-plane $T\in G(n,m)$, \begin{align*} \operatorname{div}_{T}Y_j(x)=\operatorname{div}_{T}X(\eta_{x_0,r_j}(x)). \end{align*} The mass part of the pushforward contributes the factor $r_j^{-m}$, hence the first variation satisfies \begin{align*} \delta V_j(X)=r_j^{-m}\delta V(Y_j). \end{align*} Because $V$ is stationary in $U$ and $Y_j$ is compactly supported in $U$, we have $\delta V(Y_j)=0$. Therefore \begin{align*} \delta V_j(X)=0. \end{align*} Thus on every fixed ball $B(0,R)$, the sequence has uniformly bounded mass and uniformly bounded first variation along the tail. Now Allard compactness applies on each ball $B(0,R)$ (citing a result not yet in the wiki: Allard compactness theorem for integral varifolds, including closure of integral varifolds under locally bounded first variation). Applying it on $B(0,k)$ for $k\in\mathbb{N}$ and passing to a diagonal subsequence gives an integral $m$-varifold $C$ on $\mathbb{R}^n$ such that \begin{align*} V_j \to C \end{align*} as varifolds on compact subsets of $\mathbb{R}^n$.[/guided]

[step:Pass stationarity to the varifold limit] Let \begin{align*} X: \mathbb{R}^n \to \mathbb{R}^n \end{align*} be a compactly supported smooth vector field. Since $\operatorname{spt}X$ is compact and $U_j$ exhausts $\mathbb{R}^n$, there exists $j_X\in\mathbb{N}$ such that $\operatorname{spt}X\subset U_j$ for all $j\ge j_X$. For $j\ge j_X$, define \begin{align*} Y_j: U \to \mathbb{R}^n \end{align*} by \begin{align*} Y_j(x)=r_jX(\eta_{x_0,r_j}(x)). \end{align*} Then $Y_j$ is compactly supported in $U$, and the affine chain rule for first variation gives \begin{align*} \delta V_j(X)=r_j^{-m}\delta V(Y_j)=0, \end{align*} because $V$ is stationary in $U$. Varifold convergence gives continuity of first variation against fixed compactly supported smooth vector fields with compact support in the convergence region, so \begin{align*} \delta C(X) = \lim_{j\to\infty}\delta V_j(X) = 0. \end{align*} Since $X$ was arbitrary, $C$ is stationary in $\mathbb{R}^n$. [/step]

[step:Identify the density ratios of the tangent varifold] Fix $R>0$ such that $\|C\|(\partial B(0,R))=0$. By [weak convergence](/page/Weak%20Convergence) of the mass measures at continuity sets, \begin{align*} \|C\|(B(0,R)) = \lim_{j\to\infty}\|V_j\|(B(0,R)). \end{align*} Using the scaling identity from the mass-bound step, \begin{align*} \|C\|(B(0,R)) = \lim_{j\to\infty}\omega_m R^m\Theta(V,x_0,r_jR). \end{align*} Since $r_jR\downarrow 0$, the defining limit of the density gives \begin{align*} \lim_{j\to\infty}\Theta(V,x_0,r_jR)=\Theta(V,x_0). \end{align*} Therefore \begin{align*} \Theta(C,0,R) = \frac{\|C\|(B(0,R))}{\omega_m R^m} = \Theta(V,x_0) \end{align*} for every $R>0$ with $\|C\|(\partial B(0,R))=0$. Because $C$ is stationary, the monotonicity formula for stationary varifolds applied to $C$ at the origin implies that \begin{align*} R \mapsto \Theta(C,0,R) \end{align*} is nondecreasing. The set of radii $R>0$ for which $\|C\|(\partial B(0,R))>0$ is at most countable, since the spheres $\partial B(0,R)$ are pairwise disjoint and $\|C\|$ is locally finite. Hence the monotone function $R\mapsto \Theta(C,0,R)$ agrees with the constant $\Theta(V,x_0)$ on a [dense subset](/page/Dense%20Subset) of $(0,\infty)$, and monotonicity forces \begin{align*} \Theta(C,0,R)=\Theta(V,x_0) \end{align*} for every $R>0$. [/step]

[step:Apply the equality case in monotonicity to obtain dilation invariance]Since $C$ is stationary and \begin{align*} \Theta(C,0,R)=\Theta(V,x_0) \end{align*} for every $R>0$, the density ratio of $C$ at the origin is constant in $R$. We invoke the equality case in the monotonicity formula for stationary varifolds: if a stationary $m$-varifold has constant density ratio about a point on an interval of radii, then the varifold is invariant under every dilation about that point on the corresponding annulus; in particular, if the ratio is constant for all $R>0$, then the varifold is a cone about that point (citing a result not yet in the wiki: equality case in the monotonicity formula for stationary varifolds, constant density ratio implies conical varifold). Equivalently, the equality case yields \begin{align*} x^\perp = 0 \end{align*} for $C$-almost every pair $(x,T)$ with respect to the varifold measure on $\mathbb{R}^n\times G(n,m)$, where $G(n,m)$ denotes the Grassmannian of unoriented $m$-planes in $\mathbb{R}^n$ and $x^\perp$ denotes the [orthogonal projection](/theorems/437) of $x$ onto the normal complement $T^\perp$ of the plane $T$. The cited equality case includes the rigidity conclusion that this infinitesimal radial tangency condition is equivalent, for stationary varifolds, to invariance under radial dilations. Therefore, for every $\lambda>0$, the dilation map \begin{align*} D_\lambda: \mathbb{R}^n \to \mathbb{R}^n \end{align*} defined by \begin{align*} D_\lambda(x)=\lambda x \end{align*} satisfies \begin{align*} (D_\lambda)_{\#}C=C. \end{align*} Hence $C$ is conical about the origin.[/step]

[guided]The density-ratio identity proved in the previous step is stronger than a mass estimate: it says that the monotone quantity from the stationary-varifold monotonicity formula does not increase at all. We have \begin{align*} \Theta(C,0,R)=\Theta(V,x_0) \end{align*} for every $R>0$, so the map \begin{align*} R \mapsto \Theta(C,0,R) \end{align*} is constant on $(0,\infty)$. We now apply the equality case in the monotonicity formula for stationary varifolds. The result states that if a stationary $m$-varifold has constant density ratio about a point on an interval of radii, then the varifold is invariant under every dilation about that point on the corresponding annulus; if the density ratio is constant for all positive radii, the varifold is a cone about that point (citing a result not yet in the wiki: equality case in the monotonicity formula for stationary varifolds, constant density ratio implies conical varifold). The hypotheses are satisfied here because the preceding step proved that $C$ is stationary in $\mathbb{R}^n$ and that its density ratio about $0$ is constant for every $R>0$. The same equality case can be expressed infinitesimally as follows. It gives \begin{align*} x^\perp = 0 \end{align*} for $C$-almost every pair $(x,T)$ with respect to the varifold measure on $\mathbb{R}^n\times G(n,m)$. Here $G(n,m)$ is the Grassmannian of unoriented $m$-planes in $\mathbb{R}^n$, and $x^\perp$ is the orthogonal projection of $x$ onto $T^\perp$. This says that the radial vector $x$ lies in the approximate tangent plane $T$ at $C$-almost every point. The rigidity part of the equality case is the substantive step: for stationary varifolds, this almost-everywhere radial tangency condition forces invariance under the radial flow. Thus, for every $\lambda>0$, the dilation map \begin{align*} D_\lambda: \mathbb{R}^n \to \mathbb{R}^n \end{align*} defined by \begin{align*} D_\lambda(x)=\lambda x \end{align*} satisfies \begin{align*} (D_\lambda)_{\#}C=C. \end{align*} This is exactly the definition that $C$ is conical about the origin.[/guided]

[step:Conclude existence and the stated properties of tangent cones] The compactness argument produced a subsequential varifold limit $C$ of the blow-ups $(\eta_{x_0,r_j})_{\#}V$, so at least one tangent varifold to $V$ at $x_0$ exists. The preceding steps show that every such subsequential limit is an integral $m$-varifold, is stationary in $\mathbb{R}^n$, is invariant under every dilation about the origin, and satisfies \begin{align*} \Theta(C,0,R)=\Theta(V,x_0) \end{align*} for every $R>0$. Thus every tangent varifold is a stationary integral cone with the stated density ratio, completing the proof. [/step]