The problem turned out to be a couple of things. First, the memory limit that I set was indeed too low. It needs something in the neighborhood of 2 GB of memory to boot up successfully. Second, It turns out that I hadn't mounted any storage for the notebook files.

Here's the manifest that I came up with that does work. I'm aware that the way I'm mounting storage for the notebooks is perhaps not optimal, but now that I know it's working I feel comfortable tweaking it.

---
apiVersion: v1
kind: ConfigMap
metadata:
  name: polynote-config
  namespace: dev
  labels:
    app: polynote
data:
  config.yml: |-
    listen:
      host: 0.0.0.0

    storage:
      dir: /opt/notebooks
      mounts:
        examples:
          dir: examples
---
apiVersion: extensions/v1beta1
kind: Deployment
metadata:
  name: polynote
  namespace: dev
spec:
  replicas: 1
  template:
    metadata:
      labels:
        app: polynote
    spec:
      containers:
      - name: polynote
        image: polynote/polynote:latest
        resources:
          limits:
            memory: "2000Mi"
            ephemeral-storage: "100Mi"
          requests:
            memory: "2000Mi"
            ephemeral-storage: "100Mi"
        ports:
        - containerPort: 8192
        volumeMounts:
        - name: config
          mountPath: /opt/config/config.yml
          readOnly: true
          subPath: config.yml
        - name: data
          mountPath: /opt/notebooks/
      volumes:
      - name: config
        configMap:
          defaultMode: 0600
          name: polynote-config
      - name: data
        emptyDir: {}