# 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:

- 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.
- 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 $u$.
- 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

- Solvability of (*) with Dirichlet boundary condition. This was also established by Caffarelli.
- 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.
- 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”.

- 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.

- 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.