The Hamiltonian functions $h_{t}$ along $L_{t}$ can be extended to global functions on $X$, with

$$\left\lVert h_{t}\right\rVert_{C^{0}}=O(t^{2}),\quad\left\lVert dh_{t}\right\rVert_{C^{0}}=O(t),\quad\left\lVert\nabla^{2}h_{t}\right\rVert_{L^{\infty}}\leq C.$$

But in the $t\to 0$ limit, the second derivatives fail to be continuous at the origin, and indeed $L_{t}$ changes topology in the limit.