Understanding the k-Hessian Equation and k-Curvature Equation

k-Hessian Equation

The k-Hessian equation is given by:

$H_k(u)=\sigma_k(D^2(u))=f$ (*)

where u is admissible, i.e. $\forall 1\leq i\leq k$, $\sigma_i(D^2(u))\geq 0$. This condition ensures that (*) is an elliptic equation.

The most important results are as follows:

1. Solvability of (*) with Dirichlet boundary condition. This was mainly contributed by Caffarelli in the 1990s. By using flexible functions and the maximum principle, we can establish the $C^{1,\alpha}$ estimate and $C^{2,\alpha}$ estimate in the interior. The $C^{2,\alpha}$ estimate near the boundary is established according to the conformal invariance and some perturbation of the solution of the k-Hessian equation after special rescaling.
2. Hessian measure. This is mainly the work of X.J. Wang and Trudinger. They proved that in the sense of viscosity solution, if $\sigma_k(D^2(u))=f$, then we can associate a measure $\mu$ with $u$, and the following holds: When $u\in C^2(\Omega)$, $\mu(B_r(x))=\int_{B_r(x)}\sigma_k(D^2(u))$. If $u_1, u_2, \ldots$ converge to $u$, then $\mu_1, \mu_2, \ldots$ converge to $\mu$ in a weak sense. This mainly depends on a priori estimate on $u$.
3. Pointwise estimate corresponding to Wolff potential. The Wolff potential is given by: $W^{\mu}{k}(x,r)= \int{0}^r\left(\frac{\mu(B_t(x))}{t^{n-2k}}\right)^{\frac{1}{k}}\frac{1}{t}dt$ We can easily use rescaling to understand the reasonableness of this potential. Using this potential, Lubutin established the following pointwise estimate: If $u\in \Phi_k(B_{4R}(x))$ and $u\leq 0$, then we have: $W^{\mu}k(x,\frac{R}{2})\leq |u(0)| \leq W^{\mu}_k(x,2R)-\sup{B_{2R}}|u|$ The right-hand side could be seen as a corollary of the classical A-B-P estimate, while the left-hand side needs to combine several observations, such as the mean-value property and others.

This result could be used to establish some results on singularity points that can be removable.

k-Curvature Equation

1. Solvability of (*) with Dirichlet boundary condition. This was also established by Caffarelli.
2. Curvature measure. This was established very recently. Mean curvature equation in 2014, by Perron lift and modification, and the general case in 2016 by more complex calculations and methods.
3. Pointwise estimate corresponding to Wolff potential. This is still not established and is the main focus.

My Ideas:

Looking at it as the “average” of “loop space” and “surface space”.

1. Grassmannian Bundle

In algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X:
$p:G_{d}(E)\to X$
such that the fiber
$p^{-1}(x)=G_{d}(E_{x})$
is the Grassmannian of the d-dimensional vector subspaces of $E_x$. For example,
$G_{1}(E)=\mathbb {P} (E)$
is the projective bundle of E. In the other direction, a Grassmann bundle is a special case of a (partial) flag bundle. Concretely, the Grassmann bundle can be constructed as a Quot scheme.

Like the usual Grassmannian, the Grassmann bundle comes with natural vector bundles on it; namely, there are universal or tautological subbundle S and universal quotient bundle Q that fit into
$0\to S\to p^{}E\to Q\to 0$. Specifically, if V is in the fiber p−1(x), then the fiber of S over V is V itself; thus, S has rank $r = rk(E)$ and $\wedge ^{r}S$ is the determinant line bundle. Now, by the universal property of a projective bundle, the injection $\wedge ^{r}S\to p^{}(\wedge ^{r}E)$
corresponds to the morphism over X:
$G_{d}(E)\to \mathbb {P} (\wedge ^{r}E)$,
which is nothing but a family of Plücker embeddings.

The relative tangent bundle $T Gd(E)/X$ of $Gd(E)$ is given by
$T_{G_{d}(E)/X}=\operatorname {Hom} (S,Q)=S^{}\otimes Q$, which is morally given by the second fundamental form. In particular, when d = 1, the early exact sequence tensored with the dual of S = O(-1) gives: $0\to {\mathcal {O}}{\mathbb {P} (E)}\to p^{}E\otimes {\mathcal {O}}{\mathbb {P} (E)}(1)\to T_{\mathbb {P} (E)/X}\to 0$,
which is the relative version of the Euler sequence.

2. Explanation of the Fully Nonlinear Elliptic Equation

Now, we could consider the determinant $\sum_{i_1,\ldots,i_k\in{1,\ldots,n}}\det(u_{ij}){i,j\in {i_1,\ldots,i_k}\times{i_1,\ldots,i_k}}$ as the determinant of the transform: $(u{i_1},\ldots,u_{i_k}) \longrightarrow (e_{i_1},\ldots,e_{i_k})$.

Now we need to understand $\sigma_k(D^2(u))=f$ at a point $x_0$ as the average of the determinant of the transform matrix of $(u_{i_a},\ldots,u_{i_k}) \longrightarrow (e_{i_b},\ldots,e_{i_k})$ on the Grassmannian manifold $G_k(x_0)$ is equal to $f(x_0)$, i.e.:

$\int_{G_k(x_0)} \det\left(\frac{\partial u_{i_a}}{\partial e_{i_b}}\right) d\mu=f(x_0)$

where $\mu$ is the natural Haar measure on $G_k(x_0) \simeq G_k$.

But the difficulty in making the argument rigorous is that $u_i$ is scaled and $e_i$ is a vector.